Two-coloured Path Decompositions

Two-coloured Path Decompositions

Daniel Dyer

B.Sc. (Hons), Mernorial University of Newfoundland, 1998

A THESIS SUBMI?TED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

MAsTER OF SCIENCE

In the Department

of

Mathematics and Statistics

O Daniel Dyer 2001

SIMON FRASER UNIVERSITY

Apnl2001

AU rights reserved. This work may not be reproduced in whole or in part, by photocopy

or other means, without permission of the author.

National Likaiy n * I oiC- du Cana

uisüions and Acquisitions et raphic Senricus senriees biMbgraphiques

The author has granted a non- L'auteur a accord6 une licence non exciusive licence aiiowing the exclusive permettant à la National Li- of Canada to Bibliothèque nationale du Canada de reproduce, loan, distriibute or sel1 reproduire, prêter, disûibuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de rnicroficheI'nlm, de

reproduction sur papier ou su format électronique.

The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette îhèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or othenivise de celle-ci ne doivent être imprimds reproduced without the author's ou autrement reproduits sans son permission. autorisation.

Abstract

Consider a complete graph of muitiplicity 2, where between every pair of vertices there is

one red and one blue edge. Can the edge set of such a graph be decompased into isomorphic

copies of a 2-coloured path of length 2k that contains k red and k blue edges? A necessary

condition for this to be true is n(n - 1) E O mod 2k. This is sdcient for k 5 3, and for

n 1 8k for a particular class of 2-coloured paths.

iii

Acknowledgment s

First and foremost, 1 would like to thank Kathy for her support, kindness, and particularly,

her patience. None of this would have been possible without her.

Thanks to Man, for giving me a place to hang my hat, and something to do when this

is over.

Thanks to my family and friends, for giving me a taste of civilization when 1 surrounded

myself with savages.

Finally, 1 would like to thank Tara; none of this would have been possible without her,

either.

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Acknowledgments . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . iv

ListofFigures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 TheSrnallCases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Special Cases of &k-de~~mpos i t i~n~ . . . . . . . . . . . . . . . . . . . . . . . 15

4 Colour-complementary Paths of Length 2k, k odd . . . . . . . . . . . . . . . 22

5 Lower Bound on Existence of Pzk-dt?~OmpOsitiOn~ . . . . . . . . . . . . . . . . 51

List of Figures

4.1 A path needeà to decompose 2Kz7 into colour-complementary Pis . . . . . . . 23

4.2 More paths for decomposing 2Kz7 into colour-complementary P18 . . . . . . . 24

4.3 A path needed to decompose 2K3$ into colour-complementary fZ6 . . . . . . . 26

4.4 A path needed to decompose 2K3$ into colour-complementary PZ6 . . . . . . . 27

4.5 Some paths needed to decompose 2K39 into colour-complementary P26 . . . . 28

4.6 A path needed to decompose 2 K u into colour-complementary Pz . . . . . . . 29

4.7 A path needed to decompose 2K34 into colour-complementary P22 . . . . . . . 30

4.8 Some paths needed to decompose 2K3 into colour-complementary P22 . . . . 31

4.9 A path needed to decompose 2K22 into colour-complernentary P14 . . . . . . . 32

4.10 Some paths needed to decompose 2Ka into colour-complementary P14 . . . . 33

4.11 A path needed to decompose 2Ks5 into colour-complementary fi2 . . . . . . . 34

4.12 Some paths needed to decompose 2Kjs into colour-complementary Pz . . . . 35

4.13 More paths needed to decompose 2Ks5 into colour-complementary Pn . . . . 37

4.14 Even more paths needed to decompose 2 K g j into colour-complementary Pn . 37

4.15 A path needed to decompose 2K3.5 into colour-complementary Pi+ . . . . . . 38

4.16 Some paths needed to decompose 2K3.5 into colour-complementary PL4 . . . . 39

4.17 Some paths needed to decompose 2K35 into colour-complementary PL4 . . . . 39

1.18 A path needeà to decompose 2K3s into colour-complementary P14 . . . . . . . 41

4.19 A path needed to decompose 2KJ6 into colour-complementary Pls . . . . . . . 12

4.20 Some paths needed to decompose 2KJ6 into colour-complementary Pis . . . . 43

4.21 A path needed to decompose 2KJ6 into colour-complementary Pis . . . . . . . 44

4.22 Some paths needed to decompose 2KJ6 into colour-complementary Pis . . . . 45

4.23 A path needed to decompose 2Ks6 into colour-complementary . . . . . . 16

4.24 Some paths needed to decompose 2Ks6 into colour-complementary P26 . . . . 48

5.1 A partition of the vertices of 2KZok+* . . . . . . . . . . . . . . . . . . . . . . . 52

5.2 A partition of the vertices of 2Kzok+,,, . . . . . . . . . . . . . . . . . . . . . . . 54

5.3 Two partitions of the vertices of 2Klak+, . . . . . . . . . . . . . . . . . . . . . 55

vii

Chapter 1

Introduction

Henry Dudeney (1857-1930), creator of many mathematical puzzles, posed in [4] what b e

came known as the "Nine Prisoners Problem." Given 9 prisoners, the problem was to

arrange a schedule so that for 6 days, the prisoners, chaineci in 3 distinct lines, each of three

prisoners, may take their exercise in the prison yard so that on no 2 days is the same pair

chained together. This problem is an example of what is known as a graph decomposition.

Definition 1.1 A gmph G admits an H-decomposition for some subgmph H , if the edge

set of G con be partitioned into edge-disjoint subgmphs each isomorphic to H . We Say that

G as H-decomposable, or decornposable by H .

To move the "Nine Prisoners Problemn into the laquage of graph theory, the following

definitions must 6rst be introduced.

Definition 1.2 A path oj length k is a seqttence of k + 1 distinct vertices, vi ,y, . . . ,uk+i,

joined by k edges, qu* , v2v3, ...? vkvk+l. We denote it by Pk and will wually *te it as

[ttl. 1'2. U3. . . . , u ~ + ~ ] .

Definition 1.3 A complete gmph on n vertices, denoted K,, is the graph in which evey

pair of rettices is joined by a single edge. A complete p p h on n vertices of multiplicity m?

denoted mK,, is a multigmph in which evey p i+ of vertices is joined by m edges.

So. in graph theoretical terms, a solution to the "Nie Prisoners Problem" produces a

Pi-decomposition of Kg. Of course, there is an added concern; the paths obtained by the

decomposition must be such that they c m be partitioned into 6 parts, where each vertex

CEIAPTER 1. INTRODUCTION 2

of Kg occurs in each part exactly once. This condition is known as resolvability. It is

mentioned here for completeness, but is not of concern to this thesis.

The "Nine Prisoners Problemn can be considered more generally as: For what values of

k, m, and n does a resolvable Pk-decomposition of mKn exist? This question was brought to

the fore in [6]. For k = 2, m = 1 and n = 9 (the conditions of Dudeney's original problem)

it was shown in [3] that there were 332 non-isomorphic solutions. The more general case,

only fixing k = 2, was solved in [Il.

Theorem 1.4 The gmph mKn hus a resoluable P2 decomposition if and only if n = O rnod 3

and m(n - 1) E O mod 4.

The general solution was finally given in [2].

Theorem 1.5 For k 2 2 the graph mK, kas a molvable Pk decomposition if and only il n = O m o d k + l andm(k+l ) (n - l ) sOmod2k .

Then, in [13], compiete multigaphs to complete multipartite multigraphs were consid-

ered and partial results obtained.

Definition 1.6 The complete multiprtite gmph on ni + n? + + nk vertices, denoted

mKn,,n,,...,n, is the gmph uith ils vertices prtitioned into k parts, one each of size ni,

1 5 i 5 k, where evey pair of vertices, one fmm each part, is joined by m edges and there

are no other edges. When k = 2, we caIl such gmphs bipartite.

Theorem 1.7 For k 2 3, there ezists a resoluable Pk-decomposition of T ~ K ~ ~ , ~ , , ~ ~ , . . . ,,,,

(&th ni = nr = = n, = n) if rn EO mod k + 1, mr(k + l)n n O mod 2k and either

n = O m o d k + l o r r z O m o d k + l .

In fact, slightly more than this was proven. Necessary and sufficient conditions were

given for k prime and m = 2,3.

However, the condition of resolvability may be dropped altogether, leaving the question:

For what values of k,m, n does a Pk-decomposition of mKn exist? For k < 8 this was

completely solved in [al. It was answered in [Il], with the foilowing theorem.

Theorem 1.8 The complete multigmph mKn i s decomposable into paths of length k if and

only i fn > k and mn(n - 1) = O mod 2k.

CHAPTER 1. INTROOUCTION 3

The extension of resolvable path decompositions of complete multigraphs to resolv-

able path decompositions of complete multipartite multigraphs would seem to suggest t hat

something similar might be possible when dealing with path decompositions, withoub the

condition of resolvabiiity. Even considering the simplest case, bipartite grapbs, only recently

have results been achieveà. A partial result comes in [12].

Theorem 1.9 I f a 2 b and either m as even or a and b are even, there is a Pk decomposition k + l k

of rnKalb i f and only if mab n O mod k , a 1 r-1 and b 1 r-1. 2 2

In [IO], Parker completely solved the bipartite case when m = 1, with the foliowing

theorem:

Theorem 1.10 A decomposition of Ka,* into paths of length k ezists i j and only if k[ab

and one of the folloruing conditions hold:

1. k,a, and b are even, and k 1 2a, k 5 2b, ezcept when k = 2a = 26;

2. k and a are even, b is odd, and k 5 2a - 2, k 5 2b;

3. k and b are even, a is odd, and k 5 2a, k 5 26 - 2;

4. k,a, and b are odd, and k 5 a , k 5 b;

5. k and a are odd, b is even, and k 5 a , k 2b - 1;

6. k and b are odd, a is euen, and k 5 2a - 1 , k 5 b;

7. k as odd, a and b are euen, and k 5 2a - 1, k 5 2b - 1.

This resdt has implications for the multipartite case, dthough it does not solve it

completely. For example, while K2,2 has no P3-decomposition, K2,2,2,2 does have a P3- decomposition, as shown in Figure 1.1.

A thorough discussion of path decompositions, resolvable and utherwise, is available in

[SI. The goal of this thesis is to partially answer the following question posed by M. L. Yu

in the same source.

Colour the edges of Pu, r d and blue so that there are k eàges of each colour,

and colour the edges of 2K, red and blue so that there is both a red and a blue

copy of K,. If n(n - 1) O mod 2k, is there a decomposition of the coloured

complete g a p h into copies of the coloured path?

CHAPTER 1. INTRODUCTION

Figure 1.1: A P3-decompososition of K2,2,2,*.

The condition n(n - 1) O mod 2k reflects that the number of edges of 2Kn should

be divisible by the number of edges in each path. Theorem 1.10 is very important to this

thesis, and therefore part of its proof will be provided in Chapter 3. As wellt Theorem 1.8,

will see fiequent use, though without providing a proof-

Throughout this thesis, whenever 2Kn or 2Ka,t, is discussed, it is with the assumption

that between every pair of vertices the two edges have different colours, nominally one red

and one blue.

In Chapter 2, a series of constructions determine al1 n so that 2K, can be decomposed

into al1 possible 2-coloured paths of lengths 2, 4 and 6, with qua1 nurnbers of edges of each

colour, respectively. The success of this result inspires a search for a more general solutionl

and several results that will prove useful in that search are given in Chapter 3. Chapter 4

will use these results to answer the question when the lengths of the paths, as well as the

arrangement of the colours, have special properties. Finally, Chapter 5 wiil, for 6xed k , give

a lower bound on n beyond which the question can be answered positively for all n satisfying

CHAPTER 1. INTRODUCTION 5

the necessary conditions when a particular colour condition on the paths is assumed. The

colour condition referred to is:

Definition 1.11 A 2-coloured path of length 2k on the vertices VI, v2, . . . , v2kJ v2k-ç' is

said to be CO~OU~-complementaty if for 1 i 5 2k, the edges [vit v ~ + ~ ] and [ v ~ ~ - ~ + ~ ~ u ~ ~ - ~ + ~ ] are coloured diflerently.

The fact that a lower bound exists beyond which the decomposition works is not partic-

ularly surprising. In 191, a theorem is given proving the asymptotic existence of many graph

decompcwitions, and this result is also applicable in our situation . The following definitions

are introduced, as well as a simplified version of the existence theorem. It will be discussed

further in Chapter 5.

Definition 1.12 For a vertez v of an edge-2-colound gmph H, the degree-uector of v is

r(v) = (degl(v), deg2(v)), where degi(v) is the numbet of edges of colour a incident with v.

Definition 1.13 Define a(H) to be the ieast positive integer so that (a(H),a(H)) is an

in teml linear combinatton of T ( V ) for a11 vertices v in H .

Thearem 1.14 Let H be a 2-coloured graph with m edges of each colour. There ezists a

corastant no = no(H) such that 2Kn admits an H-decomposition for al1 integers n > no such

that n(n - 1 ) = O mod 2m and n - 1 r O mod a(H).

Most of the constructions used in this thesis involve labeling the vertices of the graph

G being decomposed with the elements of a group. Then, given a permutation o of the

elements of the group, we can apply the permutation to the vertices of some subgraph H,

to obtain a new subgraph o(H) as follows. The vertices of a(H) are a(V(H)) and [u,v]

is an edge in H if and only if [u(u), ~ ( v ) ] is an edge of o(H). For example, applying the

permutation a = (1 2 3) to the path (1, 2, 31, we obtain the path [2, 3, 11. Often, we

will repeatedly apply a permutation to a subgraph. In this context, this refers to appIying

the permutation u to obtain the subgaphs H, a(H), a2(H), . . . ut-'(H) where at(H) is

the identity. In the previous example, we may apply the permutation a twice to obtain the

path a2([1, 2,3]) = [3,1,2].

Sometimes a vertex of a graph is labeled oc. This is a fixed point; applying a permutation

does not alter it. So, if the permutation (1 2 3) is applieà to the path [l, XI, 21, the

resulting path is 12, m, 31.

Many decompositions consist of a subgraph and those that result from repeatedly a p

plying a permutation to it. For example, label the vertices K4 by O, 1,2 and W. Repeatedly

applying the permutation a = (O 1 2) to the path [O, 1, oa] will produce the paths [l, 2,

oo] and [2,0, ml, the three of which form a P2decomposition of K4. We will make much use of the following simple lemma, which relates path decompositions

and colour-complementary pat h decompuaitions.

Lemma 1.15 Splitting Lemma If G zs a gmph that admils a f 2 k - d ~ c ~ m p ~ ~ i t i ~ n , then 2G admits a colour-complementary

PZk -decomposition.

PROOF. For every path in the P2k-decomposition of G, we obtain two colour-

complementary paths in 2G, in the following manner. If the uncoloured path is [v l , v2, us, . . . , u s , v ~ ~ + ~ ] , form the first colour-complementary path by colouring the edges from left to

right according to the given colour-complementary qk. The second colour-complementary

path is formed by taking the same vertices in reverse order and again colowing from left to

right. Thus, each edge is "split" into two edges, and each uncoloured path in the decompo-

sition of G gives two colour-complementary paths in the decomposition of 2G.

Throughout this thesis coloured paths of a particular type wili be written as [ui? u?, vs,

. . . , v ~ ~ ] with the colouring assumed to be written from left to right.

Chapter 2

The Small Cases

As mentioned in the Introduction, the two colours being used to colour the graphs in this

paper are red and blue. In this section, paths forming a decomposition are listed by length

and type. By type is meant the colour pattern of the edges. For instance, if the path [a, b,

c, d, e] is of type RBRB, the edges [a, bj and [c, 4 are coloured red, while [b, c] and [dl e] are

coloured blue.

As well, it shouid be noted that only non-isomorphic types are listed. So, for each type

listed, it is not possible to obtain another type in the list by the process of "fiipping" the

path or by switching the colours. The former means that both RRBBBR and RBBBRR represent the same type, and the latter that both RRBBBR and BBRRRB are the same

type.

Paths of length 2.

Up to isomorphism, there is only one 2-coloured path of length 2, and that is the path with

one red edge and one blue edge. Clearly, n(n - 1) r O mod 2 is a necessary condition for

the desired decomposition of 2Kn. But this is equivalent to n O mod 2 or n E 1 mod 2,

or that aii values of n must be considered. We denote this path by RB.

Lemma 2.1 The gmph 2K2, as decomposable into the colour-complementay path RB of

length 2 for al1 t 3 2.

PROOF. Label 2r-1 of the vertices of 2K2, with the elements of Z2r-li and the remaining

vertex m. Successively apply the permutation (O 1 2 . . . 2r - 2) to the r - 1 paths

CHAPTER 2- THE SMALL CASES 8

[- j, O, j ] , for 1 5 j 5 r - 1, and the path [- 1, oo, 11, where in each case the first edge is

red, to obtain the required decomposition. 1

Lemma 2.2 The gmph 2K2,+1 is decomposable into the colour-complementay path RB of

length 2 for al1 r 2 1.

PROOF. First label the vertices of 2K2,+1 with the elements of Z2,+1 and then succes-

sively apply the permutation (0 1 2 . . . 2r) to the r paths [-j, O, j ] , for 1 5 j 5 r.

1

Paths of length 4.

Concerning paths of length 4, we see that we have three types of paths to consider. We

must investigate two different colour-complementary paths, which we denote by RRBB and

RBRB, and the 2-coloured path RBBR. The necessary condition, n(n - 1) a O mod 4 results

in the requirement that n G 0,l mod 4.

The colour-complementary paths RRBB and RBRB

Lemma 2.3 The p p h 2K4, is decomposable into paths of types RRBB and RBRB for al1

7- 2 2.

PROOF. Label 4r - 1 of the vertices of 2K4, with the elements of Zs,-l. Label the final

vertex oc. Successively apply the permutation (O 1 2 - . . 4r - 2) to the r - 1 paths

[-1. 1 - 2j, 0, 2j - 1: 11 for 2 5 j 5 r, and the path [-2, -lt 00, 1, 21. 1

Lemma 2.4 The gmph 2K4t+r is decomposable into paths of types RRBB and RBRB for

atI r 2 1.

PROOF. Label the vertices of 2K4r+1 with the elements of Z4,+1 and successively apply

the permutation (0 1 2 . . . 4r) to the r paths [-1: -2j, 0: 2j , 11, for 1 I j 5 r . 1

The path RBBR

Lemma 2.5 The gmph 2K4, is decomposable into paths of type RBBR for al1 r > 2.

CHAPTER 2. THE SMALL CASES 9

PROOF. Label 4r - 1 of the vertices of 2K4, with the elements of Z4r-1, with the final vertex being labeled cc. Successively apply the permutation (0 1 2 . . . 4r - 2) to the

r - 1 paths [2 -2j, O, 2j - 1, 1, 2j], for al1 2 5 j 5 r? and the path [O, oo, 1, 2, 31. 1

Lemma 2.6 The gmph 2K4,+1 is decomposable into paths of type RBBR for al1 r >_ 1.

PROOF. Label the vertices of 2K4r+l with the elements of Successively apply the

permutation (0 1 2 . . . 47) to the r paths [l - 2 j , 0, 2j, 1, 2 j + 11, for al1 1 5 j 5 r . rn Combining the results of t his section, we obtain the following theorem.

Theorem 2.7 The gmph 2K, is decomposable into a given 2-coloured path of length 4 i f

and only if n 2 5 and n(n - 1) r O mod 4.

Paths of length 6.

Here, we have seven different path types to consider. They are RRRBBB, RBRBRB, RRBRBB, RBBRRB, RRBBBR, RBRBBR and RRBBRB. This time, the necessary condi-

tion n(n - 1) = O mod 6 implies that n E 0,1,3,4 mod 6. We wili make use of the following

definition.

Definition 2.8 The length of an edge [a? b] in K,, whose vertices are the elements of Zn is n

la - 61. Thus edge lengths are 1, 2, . . . , and LZj.

The colow-complementary paths RRRBBB and RBRBRB

Lemma 2.9 The gmph 2K6, is decomposable into paths of type RRRBBB and puths of type

RBRBRB for al1 r 1 2.

PROOF. Label 6r - 1 of vertices of 2Ks, with the elements of Z6,-~. Label the remaining

vertex w, a fixeci point, Successively apply the permutation (0 1 2 . , . 6r - 2) to the

r - 1 p a t k (2 - 3j, -1, 1 - 3j, 0, 3j - 1, 1, 3 j - 21, for 2 5 j < r! and the path [-2, -3,

-1, 30, 1, 3, 21. 1

Lernma 2.10 The gmph 2K6r+i as decomposable into paths of type RRRBBB and paths of

type RBRBRB for al1 r 2 1.


PROOF. Label the ôr + 1 vertices of 2Kst+~ with the elements of &+l. Successively

apply the permutation (0 1 2 . .. 6r) to the r paths [l - 3j, -1, -3j, 0, 3j, 1, 3 j - 11, for 15 j sr.

Lemma 2.11 The graph 2K6r+3 is decomposable into patlis of type RRRBBB and paths 01 type RBRBRB for al1 r 2 1.

PROOF. Label the 6r + 3 vertices of 2K6r+3 with the elements of Zsr+3. Successively

apply the permutation (0 1 2 . . . 6r + 2) to the r paths [-3j7 -1, -3j - 1, 0, 33' + 1,

1, 3j], for i 5 j < r. This wiii use al1 edges of 2Ksri3 except those of length 1.

To finish, successively apply the permutation (O 3 6 . 6r)(l 4 7 -. - 6r + 1)

(2 5 8 .-. 6r+2) to thepath [O, 1, 2, 3, 4, 5, 61.

Lemma 2.12 The gmph 2K6r+4 is decomposable into paths of type RRRBBB and paths of

type RBRBRB for al1 r 2 1.

PROOF. Label 6r + 3 vertices of 2K6,+4 with the elements of Z6r+3; the remaining

element wiii be labeled oo. Successively apply the permutation (O 1 2 . 6r + 2) to

the r - 1 paths [-33, -1, -3j - 1,0, 3 j + 1, 1, 33'3, for 2 5 j 5 r! and the path [-2, 5, -1.

ml 1, 5, 21. The edges remaining wili be those of lengths 1 and 2. These are used by sucessively

applying the permutation (O 3 6 . . . 6r)(1 4 7 - - 6r + 1)(2 5 8 - . 6r + 2)

to the two paths [O, 1, 2, 3, 4, 5, 61 and [O, 2, 4,6, 8, 10, 121. 1

The colour-complementary path RRBRBB

For decompositions into paths of type RRBRBB, we can use the constructions of Lemma 2.9

and Lemma 2.10 for the graphs 2Ksr and 2K6,+i, respectively.

Lemma 2.13 The graph 2K6,+3 ts decomposable into path of type RRBRBB for al1 r 1 1.

PROOF. For 2K9, we need only examine the underlying graph Kg. B y Theorem 1.8, K9

is decomposable into paths of length 6. By Lemma 1.15, the result foilows.

For r 2 2, Iabel the vertices of 2K6,+3 with the elements of &+3. Then successively

apply the permutation (0 1 2 -. . 6r + 2) to the r - 1 paths [3-3j, -1,2-3j, 0, 3j-2,

1, 3j-31, for 2 5 jsr.


To obtain the remaining edges of 2K6,+3, successively apply the permutation

(O 3 6 6r ) ( 1 4 7 .-• 6r +1) ( 2 5 8 - - . 6 r + 2 ) to the four paths [-3,

-3r - 2, -1, 0 , 3r, 1, 1 - 3r], [-2, -3r - 1, 0, 1, 3r + 1, 2, 1 - 3r], [-1, -3r, 1, 2, 3r + 2,

3, 2 - 3r] and [O, 1, 2, 3r + 2! 3r +3? 3, 3r+4] . 1

Lemma 2.14 The graph 2K6r+l is decomposable into paths of type RRBRBB for al1 r 2 1.

PROOF. For 2Kio, label 9 of the vertices with the elements of &, and the remaining ver-

tex m. Then successively apply the permutation (0 3 6 . . . 6r)( l 4 7 . . . 67 + 1 )

(2 5 8 6 r i 2 ) to thepaths [O, 1,2,m, 5 ,4 ,3] , [l, 5 , O , m , 3 , 8 , 4 ] a n d [2,0! 7 ,

CQ, 4, 6, 81. The following paths use the temaining edges of 2Klo [6, 0 , 8, 2, 4, 7, 31, [3, 7 ,

4, 2 ,8 ,0 , 61, [3, 6 , 5, 8, 1,4, 01, [O, 4, 1, 8, 5, 6,3], [O, 3, 2, 5, 7, 1, 61 and [6, 1, 7 , 5, 2, &O]. For 2K6r+r, with r 1 2, label 6r + 3 of the vertices with the elements of 26r+3, and

the remaining vertex a. Successively apply the permutation ( O 1 2 +. . 6r + 2) to the

paths [2 - 3j, -1,l - 3j,0,3j - 1,1,3j - 21, for 2 < j 5 r , and [-2, -3r - 2, - 1 , q 1,3r + 2,2], and then successively apply the permutation (0 3 6 . -. 6r)( l 4 7 . -. 6r + 1)

(2 5 8 - - - 6r+2) to thepaths[O, 1 ,2 ,3 ,5 ,7 ,9] and [9 ,7 ,5 ,3 , 2, 1,0]. 1

The colour-complementary path RBBRRB

Once again, we may use the constructions of Lemma 2.9 and Lemma 2.10 when considering

2Ksr and 2K6r+i. We may aIso use the construction of Lemma 2.14 when considering

2K6,++ This leaves the case 2K6,+3.

Lemma 2.15 The graph 2K6r+3 zs decamposable into paths of type RBBRRB for al1 r 2 1.

PROOF. We know that Ks admits a P6-decomposition (by Theorem 1.8). Using

Lernma 1.15, we have the desired decomposition of 2Kg. Consider 2K6r+3, with r 2 2, and Iabel the vertices with the eiements of Then

successively apply the permutation (0 1 2 . - - 6r + 2) to the r - 1 paths [-3 j, - 1, -3 j - 1,01 3 j + 1,1,3j], for 2 5 j < r , and successively apply the permutation ( O 3 6 - - 6r)

( 1 4 7 - - - 6r+1)(2 5 8 ... 6r+2)tothefourpaths[O,1,2,4,5,6,7],[-3, -1,

-4,0,4, 1,3], [-2,0, -3, 1 ,5 ,2 , 31 and [-1, 1, -2,2,6, 3, 51. 1

CHAPTER 2. THE SMAJX CASES 12

The path RRBBBR

Lemma 2.16 The gmph 2K6, is decomposable into paths or type RRBBBR for al1 r 2 2.

PROOF. Label 6r - 1 of the vertices of 2K6, with the elements of Zsr-I. Label the

remaining vertex m. Successively apply the permutation (O 1 2 . . . 6r - 2) to the

r - 1 paths (-1, 2 - 3 j , 0, 3 j - 1, 2 ,3 j , II, for 2 5 j 5 r and to the path [-1, -2, m, 2, 1,

3, 51. 1

Lemma 2.17 The gmph 2K6,+, is decomposable into paths of type RRBBBR for al1 r 2 1.

PROOF. Label the 6r + 1 vertices of 2K6,+~ with the elements of Z~,+I. Successively

apply the permutation (0 1 2 . . . 6r) to the r paths [-1, -3j + 1, 0 , 3 j , 2, 3 j + 1, 11,

for 1 5 j s r .

Lemma 2.18 The gmph 2K6,+3 is decomposable into puths of type RRBBBR for al1 r 2 1.

PROOF. Let the vertices of 2Ks,+3 be labeled with the elements of Z6r+3. Then suc-

cessively apply the permutation O 1 2 . . . 6r + 2 to the r paths [ - I l -3j, 0, 3 j + 1, 2,

3 j + 2 , 11, for 1 < j 5 r , and thepennutation(0 3 6 a - . 6r) (1 4 7 a . - 6 r + 1 )

(2 5 8 .-. 6 r + 2 ) tothepath[O, 1, 2.3,4, $61. 1

Lemma 2.19 The graph 2K6r+4 is decomposable into paths of type RRBBBR for all r 2 1.

PROOF. Let 6r+3 of the vertices of 2K6r+4 be labeled with the elements of Z6r+3 - the final vertex being labeled 0x1. Then, successively apply the permutation (0 l 2 . - - 6r + 2)

to the r - 1 paths [-1, -3j + 2, 0, 3j - 1, 2, 3 j , 11, for 2 5 j 5 r and the path [-3r, 0,

00, 1, 3r + 2, 2, 1 - 3r]. Also, successively apply the permutation (O 3 6 - . . 6r)

( 1 4 7 - S . 6 r f 1 ) ( 2 5 8 ... 6 r + 2 ) to the paths [O! 1, 2, 3, 4, 5, 61 and [O, 2. 4.

6, 8, 107 121. 1

The paths RBRBBR and RRBBRB

Lemma 2.20 The graph 2K6, is decompsabfe into potlis of type RBRBBR and paths of

type RRBBRB for al1 r > 2.


Lemma 2.23 The gmph 2K6r+4 is decomposable into paths of type RBRBBR and paths of

type RRBBRB for al1 r > 1.

PROOF. For 2Kio, label nine of the vertices with the elements of Zg and the remaining

vertex W. Successively apply the permutation (0 1 2 . - - 8) to the 2-coloured path [5,

O, m, 2, 6, 8, 11. Also, take thepaths [O, 3, 2, 5, 4, 7, 61, [6, 0, 8, 2, 1, 4, 31, [3, 6, 5, 8, 7, 1,

O], (4, 5, 2, 3, 0 , 1, 71, (7, 8, 5, 6, 3, 4, 11 and [ L 2, 8, 0, 6, 7,41.

For r 2 2, label 6r + 3 of the vertices of 2Ks,+4 with the elements of Z6r+3, and the

remaining vertex oo. Then successively apply the permutation (0 1 2 . . . 6r + 2) to the

r - l p a t h s [ - 1 , - 3 r + 3 - 4 , 0 , 3 r - 3 j f 4 , 1, -3r+3j-1 , -6r+6j-31,for 1 5 j s r - 1

and the path [-4, 0, w, 2, 6, 9, 121.

Finally, we also need the 2k + 1 paths: [O, 1, 2, 3, 4, 5, 61, [6, 7, 8, 9, 10, 11, 121,. . . , [6k - 6, 6k - 5, 6k - 4, 6k - 3, 6k - 2, 6k - 1, 6k] , [6k, 6k + 1, 6k + 2, 0, 2, 4, 61, [6, 8, 10,

12, 14, 16, 181,. . . , [6k - 9, 6k - 7, 6k - 5, 6k - 3, 6k - 1, 6k + 1.01 and [4, 3, 2, 1 , 0 ,6k + 1,

6k - 11, [6k - 1, 6k - 3, 6k - 5, 6k - 7 , 6k - 9, 6k - 11, 6k - 131, . . . , [14, 12, 10, 8, 6, 4, 21,

[2, 0, 6k + 2, 6k + 1, 6k , 6k - 1, 6k - 21,. . . , and [ I O , 9, 8, 7: 6 , 5, 41.

Combining the results of this section, we obtain the following theorem.

Theorem 2.24 The complete gmph 2K, is dewmposable into 2-coloured paths of length 6

if and only i f n > 6 and n(n - 1) s O mod 6.

Having found that, without exception, whenever k 5 3 and n(n - 1) z O mod 2k, that

2K, can be decomposed into 2-coloured paths of length 2k suggests that the result may

hold for al1 k. h t h e r , when k = 3, it is clear from the above lemmas that decompositions

were much easier to formulate when the paths were colour-complementary. By focusing

on colour-complementary paths, we can extend this result to larger k , as will be shown in

Chapter 3.

Chapter 3

Special Cases of P2k-decompositions

In order to eventually tackle the general problem of decomposing 2Kn into isomorphic

2-coloured paths, we will first examine some special cases. In particular, we examine con-

structions for decomposing 2K2k (for r > 1) and 2K2kr+l into coloured paths of length 2k

for al1 r. These decompositions wili also yield a nice corollary concerning paths of length

2m+1. interestingly, these parameten are the same as thme of the early steps in solving the

cycle decomposition problem [l] .

Complete Graphs

Both of the proofs of the following theorems describe constructions for decomposing partic-

ular complete graphs of multiplicity 2 into colour-complementary P2&.

Theorem 3.1 The gmph 2K2kr+l is decomposable into isomorphic copies of any fized

colour-complementary path of length 2k, for al1 positive integers r and k.

PROOF. Let the vertices of 2K2k,+i be labeled with the elements of & k t + ~ . We wili

define r paths to which the permutation (O 1 2 . - . 2kr) will be successively applied.

This construction wili be dependent on the parity of k.

If k is even, the paths are k 1-4 - jk-1+$, - 5 ~ 1 , - jk -2+T .... , -3. -jk+2, -2, - j k + l , -1, -jk,O, jk,

k 1. jk-1 .2 . jk-2 .3 ,..., j k - 5 + 2 , f - 1 , j k - ! + l , $ ] , w h e r e l < j < r .

If k is odd, the paths are

i-jk + y, -9, -jk - 1 + y, -9 + 1, ...' -jk + 1, -2, -jk + 1, -1, -jk, O, '-' 1, jk - 9 + 1, y, jk - y], where 1 < j < r. jk. 1. jk -1 .2 , jk-2 ,..., -- 2

CHAPTER 3. SPECIAL CASES OF P2k-DECOMPOSITIONS 16

The basic idea behind this construction is that edges of al1 lengths are used exactly

twice, and in particuiar, if e and f are twa edges equidistant from the beginning and end

of a path, then e and f have the same length. Since the paths are colour complementary,

e and f are coloureà differently. Therefore al1 lengths of both colours are represented. By

applying the permutation, al1 edges of these lengths are used.

Theorem 3.2 The gmph 2Kzk,, is decomposable into isomorphic copies of anyfized colour-

complementa y p t h of length 2k for al1 positive integers k, and t > 1.

PROOF. Label 2kr - 1 of the vertices of 2Kzk, with the elements d Z2kr-i and the remaining

vertex m. Successively apply the permutation (O 1 2 . . . 3kt - 2) to the given base

paths. Once again, there wiU be two cases depending on the parity of k. k If k is even, the paths are [ - 5 , - j k + k, - 5 + 1, - jk + 9 - 1, . .. , -3, -jk + 3, -2,

k k - j k + 2 , -1, - j k+1,0 , jk-1, 1, j k - 2 , 2 , j k - 3 , 3 , ..., j k - 2 + 1 ! 2 - 1. j k - 5 , $1, k f o r l c j < r , a n d t h e p a t h [ - ~ , - ~ , - ~ - 1 , ~ + 1 , ..., -k+2, -3 , -b+l , -2 , -k ,

k -1.00, 1, k.2, k -1 .3 . k - 2 , ... 5 - 1, $ + 2 , i, 1+1]. k+l 1f k is odd, the paths are [-jk+ 9, -y, -jk+ T + 1, -y + 1. .. . , -2. -jk+2,

k+L k-1 .k 1, jk--2+1, 7 , ~ -y]. for 1 < j s r , -1: - jk+1,0, jk-1, 1 , jk-2 ,2 , ..., m- k+L kt3 k c l k+3 and the path [ -2 , -T, -2 + 1, -F - 1, . . . , -k + 2, -3, -k + 1, -2, -k, -1, x,

k+3 k+l k+3 k+t 1, k, 2, k - 1: 3, k - 2, - . - 3 -2 + 1, 2- - 1, '2,

Again. these paths are constructeci so that edges equidistant from the endpoints of the

path have the same length, and hence receive dierent colours.

Coroilary 3.3 Let k = 2m, with m a positive tnteger. Then 2K, is decomposable into

isomorphic copies oJ any fized colour-complementary path of length 2k i f and only if n > 2k

and n(n - 1) r O mod 2k.

PROOF. Since n(n - 1) E O mod 2k and k = 2'", it follows that n E O: 1 mod 2"'+'.

Equivalently n = 2k7 or n = 2kr + 1 for some r. and the decomposition is provided by

Tbeorems 3.1 and 3.2.

However, this particular case, while a simple consequence of the two previous theorems,

does seem to imply that decomposing 2K, into colour-complementary paths of certain

lengths may be easy to achieve, motivating the subsequent investigations of this thesis.

CHAPTER 3. SPECIAL CASES OF &-DECOMPOSITIONS

Path decompositions of bipartite graphs

The following section is the work of C. Parker and comes from her Ph.D. thesis [IO]! with

only minor structural changes. It is included here for completeness. In fact, this is the proof

of only a small part of the proof of Theorern 1.10, but these are the only results that we

wiil use.

The graphs in this section are not multigraphs! and the paths are not colour-

complementary, or even coloured. However, these decompositions wiil play an important

part in decomposing muitigraphs and in the recursive constructions of Chapters 4 and 5.

We begin with some definitions.

a Definition 3.4 Define Calb to be the multi-cycle with b uertices and edge-multiplicity 1-J,

2 in which

1. the b uertices are labelai by ui, i E 5, and

2. the edges fima vi to v,+i a n lokied 4 for O < j < ig j - 1. We think of these edges as k ing oniered from the "inside" to the "outside" of the cycle,

with a! as the inner-most edge from V i to Vi+i.

a Definition 3.5 Define P ( i ) , i E 5, to be the path in Calb of length 1-1 that begins ut V i

2 LIJ-1 1 3 - 2 l:l-3, - . , and and uses the vertices ui,vi+i,ui+l,. . . , ui+[;] and the edges ai , a i f l , ai,*

O a i + [ ~ ~ - l .

Lemma 3.6 Let a and b & positive even integers. If there exists a proper colouring of the

edges of Co,b using a colours such that the edges of P(i ) receive distinct colours for each

i E Zb, then there exists a decomposition of Ka,* tnto b paths of length a.

PROOF. Assume that such a colouring of Ca,b exists. Label a set of vertices of size a

(call it A) with the a colours. Label another set of vertices of size b (call it 8) with the

elements of V(COTb). An edge exists between an element i of A and an element vj of B if there is an edge of colour i incident with V j in The resulting graph is clearly bipartite.

Since a is even, by definition there are edges between v,-1 and ui in Ca,b. Siniilarly

for vi and vi+l, making for a total of a edges incident with ui. However, as the colouring is

proper, each of the a colours must occur exactly once. Thus, each element of A is adjacent

to each element of B exactly once, and the graph formed is

CHAPTER 3. SPECXAL CASES OF P2k-DECOMPOSITIONS 18

Now, consider a path P(i) in Ca,b* Let ah be an edge in this path of colou. j; that

is, an edge between the vertices v,,, and u,+i. This corresponds to the path [v,, j, v,+~]

of length 2 in Kalb. Since each eàge in P(i) receives a Merent colour, the corresponding

subgraph in Kalb is a path, P'(i). Moreover, as P(i) is of length ;, the path P(i) is of

length a.

Finaliy, consider an edge [i, vil in Kala. This means that one of the edges between uj- 1

and vj or between vj and uj+i has colour i. This i-coloured edge wiil occur in exactly one

path P(m) in Ca,b; hence [t, v j ] will occur in exactly one path E)'(m) in Ka,b. Thus this

construction ensures that every edge in Kala occurs in exactly one path of length a, giving

a decomposition of Ka,b into paths of length a.

Theorem 3.7 If a and b are even integers and a 5 26, then Ka,b as decomposable into pths

of length a .

PROOF. Colour the graph Calb by the following rule. The edge 4; (the jth edge between

vi and vi+,) is assignecl colour 2 j if i is even, for O < j 5 8 - 1, and colour 2 j + 1 if i is odd,

for O 5 j 5 4 - 1. By construction, this colouring is proper, as al1 edges between oi and

vi+l will be coloured with colous of dinerent parity than the edges between vi-1 and ui.

Now, consider a path P(i ) . If i is even, then the sequence of colours corresponding to

the edges of the path [vi, ui+l, vi+*, . . . , ui+;] will be a - 2, a - 3, a - 6, a - 7 , . . . ,1! 3,

0, respectively. The colour of each edge differs fiom the preceding by at most two. Since

there are only 8 edges, the fint and iast colour differ by at most a. In fact, since no edges

differ fkom another by 2, al1 must be colourd distinctly. Similarly for odd i. Thus, since

we have a proper colouring of the edges of Co,b which leaves ali paths P(i) with distinct

colours, Kata is decomposable into paths of length a by Lemma 3.6. 1

Lemma 3.8 Let a and b be positive integers unth a odd. I f there exists a colouring of the

edges and vertices of Ca,b vsing a colours, svch that

1. each colour appears at each uertez ezactly once, and

2. the edges and final vertez of P ( i ) receiue distinct colours for al1 i E Zb!

then them ezists a decomposition of Ka,b into paths of length a.

PROOF. Assume that the edges and vertices of Calb are coloured as describecl. As before

consider a set A of size a, the eIements being the colours used, and a set B of size b, the

CHAPTER 3. SPECIAL CASES OF Pzk-DECOMPOSITIONS 19

elements being the vertices of Cala. Draw edges between an element i of A and an element

vj of B if there is an edge coloured i incident with vj in Cab, or if the vertex vj has colour

i. Since each colour appears at each vertex exactly once, we know that the resulting graph

is

Consider the edges of the path P(i) (of length y). Since the colour of each edge is

difïerent, we obtain a path of length a - 1 in Ka,b, as in the previous lemma. However, as

the final vertex in P(i) also bas a different cotour, this induces a single edge in Ka,b to a

vertex in A not previously used. This edge, combineci with the path of length a - 1 produces

a path P ( i ) of length a.

Now, let [i, v j ] be an edge in Wit h the assumed colouring, either exactly one edge

incident with v, has colour i or the vertex uj has colour i . If the former, the i-coloured

edge is used in exactly one P(m), and hence [i, vj] is in exactly one Pt(m), as before. If the

latter, the vertex v, is used as a final vertex in exactly one P(m) (by the construction of

the P(m)), and hence [i, vj J appears in exactly one P(m). Thus, every edge in Ka,b is used

in exactly one path in the given construction, proving that we have a decomposition of Ka,*

into paths of length a. I

Theorem 3.9 If a is an even integer, b is an oàd integer and a < 6, then Ka,b is decom-

posable into paths of lenglh a.

PROOF. We will first form a decomposition of into paths of length a - 1. Cet

a' = a - 1. Colour the edges of the graph by the following rule: an edge ai gets colour

i -i 2j + 2 mod a' for O < j < 9. As WU, d o u r vertex vi with colour i.

Now, replace vertex uo with b - a' + 1 vertices, labeled U O , ~ , O 5 k < b - a'. The vertex

v,,,~ will be adjacent to vo,~, with edges coI.s?id the same as those in C,+,,l between w-1

and vo; similarly for vo,b-.~ and VI. Join vo,i and vo,i+~ with 9 edges, with edges being

labeled a;,,. Colour al1 the v0.i with colour O. Colour the edges by the following rule, for

O < j 5 b - a': if k is even, edge a& wiil receiw colour 2j + 2 and if k is odd, edge d,,, wili

receive colour 2j + 1. Consider vertex v,, with i # O. In this case? the vertex receives colour i, and the colour

of the edges between vi-1 and vi are of dinerent parity than the edges between vi and Vi+l,

and by construction, ali are distinct. Thus, each colour appears at each vertex exactly once.

A similar argument holds for the vertices Vo,i-

CHAPTER 3. SPECZAL CASES OF PZk -DECOMPOSZTZONS 20

We now need only consider the paths P(i) (or P(0, i ) , defined similarly). First , consider

a path P(i) which contains only vertices of type v,. Such a path would have edges which

have the following colours: i + a' - 1, i + a' - 2, i + a' - 3, . . . , i + y, with the finai vertex

having colour i + 9, ail of which are distinct.

A path P(0, i ) that contains only vertices of type V o j would have edges with the following

colours if i is even: a' - 1, a' - 4, a' - 5, a' - 8, a' - 9, . . . , 5. 4, 1, with the final vertex

being coloured O. Note that every two edges have their colours decrease by 4. Their are

9 edges, and so 9 pairs of edges. This makes for a maximum decrease of a' - 1, and

hence al1 of these colours are distinct.

Similady, when i is odd, P(0, i) will have edges coloured a' - 2, a' - 3, a' - 6, a' - 7,

d - 10, . . . ,6,3,2, with the final vertex again being coloured O. The same argument suffices

to show that no colour is repeated.

FinalIy, we must consider paths that contain vertices of both types; that is, paths that

cont ain vertices of the form vi and vo ,j . By our colouring of the edges we know t hat colours

of the edges of the iorm [vi, v ~ + ~ ] difter by at most 1 from the preceding edge (as do edges

of the form [vi, vo ,j] and [vojY vil). By the colouring rule given for the vertices that replace

vo, we know that the colours of edges of the form [vOti,vo,i+ll can d8er by at most 3 from

those preceding of foilowing the edge. However, every edge that does differ by 3 from an

edge preceding or following it ddfers only by 1 from the other edge incident with it. As well,

the colour of the h a l vertex can only differ by 1 or 2. We must show that al1 the colours

are distinct. But this is the same as showing that the range of colours is l e s than a. So, we

must consider the case when we use as many edges as possible with a colour that difTers by

3 from the colour of the preceding edge as possible. But this is exactly the previous case.

and hence aii the colours are distinct.

So, every colour appears at every vertex of the multi-cycle CalTb exactly once, and the

paths P( i ) all receive distinct colours on their edges and final vertex. So, by the previous

lemma, there is a decomposition of Kat,b into paths of length a'. In fact, we have a Little

more. By the method described in Lemma 3.8, there are b paths, each with a different

endpoint in the bset of Katlb- Now, consider again the graph Ka,*. Pick some point u in the a-set of the vertex

partition. Ignoring that vertex, we can decompose the graph Ka-r,b into paths of length

a - 1, each of which ends on a ddferent one of the vertices of the bset. For each path so

forrned, add the edge from v to the endpoint in the bset, forming a path of length a. and

CHAPTER 3. SPECIAL CASES OF P2k-DECOMPOSITIONS

hence a decomposition of Kalb into paths of length a, as required.

Combining Theorems 3.7 and 3.9, we easily obtain the following corollary.

Coroiiary 3.10 The gmph c m be decomposed into paths of length 2k for al1

positive integers m.

Note that the simplet case, when m = O c m be solved much more easily by a direct

construction.

Theorem 3.11 The graph K2k,2k can be decomposed into paths of length 2k.

PROOF. Label the vertices of K2k2k with the elements (1,0)? (1, l), (1,2),. . . , (1,2k - l), (2,0), (2, l), (2,2), . . . , (2,2k - 1). Then the path [(1, k), (2, k), (1, k - l), (2, k + 1):

(1, k - 2), (2, k + 2), . . . , (2,2k - 1),(1, l), (2,2k - l), (1,0)], dong with the paths obtained

£rom repeatedly applying the permutation (O 1 2 . . . 2k - 1) to the second coordinates

of the vertices of this path. B

Chapter 4

Colour-complementary Pat hs of

Length 2k, k odd

We have already seen that when considering colour-complementary paths of length 2k =

2m+1. al1 appmpriate complete graphs admit a path decomposition. This result suggests

something further; namely, that paths of "nice" lengths may have some useful properties.

In fact, this is exactly the case when dealing witb paths of lengt h 2k, with k an odd integer.

Lemma 4.1 For udd k , the gmph 2K3k is decomposable into isomorphic copies of any b e d colour-complementary p t h o j length 2k.

P ROOF. First, note that if k 3 mod 4, then 1 E(Ku)I is divisible by 2k. Thus, K3k is decomposable into paths of length 2k by Theorem 1.& and by Lemma 1.15: we obtain a

CHAPTER 4. COLOUR-COMPLEMENTARY PATHS OF LENGTH 2k, k ODD 23

as iiiustrated in Figure 4.1 when g = 1.

Figure 4.1: A path needed to decompose 2KZ7 into colour-complementary Pia.

We aiso consider the two additional paths formed by (repeatedly) applying the permu-

tation (1 2 3) to the fint coordinates of the vertices of this path. Similady, we repeatedly

apply the permutation (0 1 . . . 89) to the second coordinates of the vertices of the three paths formed above. As the k t permutation has order 3, and the second has order 8g + 1,

we have now used 3(8g + l)(16g + 2) edges. We denote the paths that result from this

construction P-paths. These P-paths use aii of the edges in the three copies of 2K8g+l,

and al1 the edges of lengths O, 1, . . . , 29 and 6g + 1, 6g + 2? . . . , 89 in the three gaphs

2K89+1,89il- However, more paths are needed to complete this construction. For each à such that

O 5 i < g - 1, consider the foilowing uncoloured paths of length 16g + 2: [(l,O), (2,4g-2i), (1,1), (2,4g+1-2i), (1,2), ... (2,8g-2), (1?4g-1+2i), (2,8g-1).

(1,4g + 24, (2, Sg), (1,4g + I + 24, (2,O): (1,4g + 2 + 2i), (2, l), (1,4g + 3 + Z i ) , . . . (2,4g - 3 - 2i)? (1,8g - l), (2,4g - 2 - 2i), (1,8g), (2, 4g - 1 - 2i), (3,O)I

and


[(3,0), (2,4g-2i), (3,1) , (2,4g+1-2i), (3,2) , ... (2,8g-2), (3,4g-1+2i), (2 ,8g- l ) ,

(3-49 + 29 , (2,8g), (3,4g + 1 + S i ) , (2, O) , (3,4g + 2 + 2i) , (2, l ) , (3,4g + 3 + 2i), . . . (2,4g - 3 - 2 9 , (3, Bg - l ) , (2,4g - 2 - 2i), (3,8g)? (2,49 - 1 - 24, (1, O ) ] , as illustratecl in

Figure 4.2 when g = 1 and i = 0.

Figure 4.2: More paths for decomposing 2Kz7 into colour-complementary PlB.

These two paths each correspond to two colour-complementary paths, on applying

Lemma 1.15. As well, the permutation (1 2 3) should be repeatedly applied to the first

coordinates of the vertices of each of the resulting colow complementary paths. We cal1 these paths Q-paths. Sol for each of the g possible values of i! we obtain 12 different paths.

So, Q-paths contribute a further 12g(16g + 2) edges of 2K3(89+1). They also use al1 the

edges of lengths 29 + 1, 29 + 2, . . . , and 6g in each of the three copies of 2K8g+1,89+ 1. In

total, the nurnber of edges this construction uses is


Now, to prove the correctness of this construction, it is oniy needed to show that every

edge is used at least once. Without 105s of generality, consider a red eàge e = [(a, b), (c, d)] .

Fust, consider a = c. Let 6 E b - d mod 49, O 5: 6 5 49 - 1. The P-paths above use

two edges of length 6, one of each colour. By applying the permutation (1 2 3), the path

may "shifted" to one containhg edges of the type [(a, b), (a, d)]. Then, since there are only

89 + 1 edges of length 6 with k t cooràinate of each vertex a, we must use al1 of them on

repeatedly applying the permutation (O 1 - . . 89) to the first coordinates. Thus al1 such

eàges are useà.

If a # c, we need to consider two cases. Define b = b - d mod 89, O < b 5 89 - 1. It can

be seen t hat if eit her O 5 6 5 2g or 6g < 6 5 89 - 1, then e is merely the image of an edge of

a P-path, under some permutation (1 2 3)'(0 1 . . . 8g)s, with O < t 5 2, O < s 5 8g.

If 6 is instead between 29 + 1 and 69 - 1, we consider one of the Q-paths, as they (or

their images under the permutation (1 2 3)) use aii such edges. In particular, edges of

lengt hs 49 - 2 j and 49 - 2 j - 1 are used when i = j. Thus, every edge occurs in some pat h

in the construction. Since ewry edge occurs, and the construction oniy uses 1 E(2K3(89+1))1

eàges, every edge must be uaed exactly once, and hence this is a valid construction.

We must also consider how to decompose 2K3k when k = 89 + 5, for al1 non-negative g.

When g = O, that is, when we are interested in decornposing 2Kt5 into paths of length 10,

label the vertices of 2KI5 m follows: (1, O), (1, l ) , . . . , (1,4), (2, O), (2, l), . . . , (2,4), (3, O),

(3, l ) , . . . , (3,4). Then consider the following path:

[ ( b l ) , (113), (3,3), ( h 4 h (2,3), (2,% RU, (3,OL (2,4), ( 3 , 4 ? (3,211- Also consider the paths formed by repeatedly applying the permutation (1 2 3) to

the first coordinates of the vertices, and the permutation (O 1 2 3 4) to the second

coordinates of the vertices. This produces 15 colour-complementary paths.

We must also consider the following uncoloured path: [(l, O), (2,3), (1, l) , (2, 4): (1,2),

(3, O)? (2, 21, (1?4), (2711, (1931, (2, O)]. As before, repeatedly apply the permutation (1 2 3) to the first coordinates of the

vertices of this path to obtain two more paths. Then colour them according to Lemma 1.15.

This completes the 2Ki5 decompcwition.

To decompose 2K3(89+53 into colo~~-complementary paths of length 16g + 10, when

g > O, label the vertices of 2K3(sg+s) by (1, O), ( l , l ) , (1,2),. . . ,(l , 89 + 4), (2, O), (2, l ) ,

(2,2),. . . , (2, 89 + 4), (3, O), (3, l), (3,2), . . . , (3,8g + 4). Now, consider the following path:


Figure 4.3: A path needed to decompose 2KS9 into colour-complementary PZ6.

Of course, as before, we m u t also consider the paths formeci by repeatedly apply-

ing the permutation (1 2 3) to the first coordinates of the vertices, and the permu-

tation (O 1 - - - 89 + 4) to the second coordinates. This .produces 3(8g + 5) colour-

complementary paths, which we wüi cal1 P-paths. Further the P-paths uses aii the edges of


Figure 4.4: A path needed to decompose 2K39 into colour-complementary qs.

and

[(1, O), (2,49 + 1 - Si), (1, 1): (2,4g + 2 - 2i), (1,2),. . . (1,49 + 2 + 2i), (2,8g + 3),

(1,49 +3+2i), (2,89+4)? (1,4g+4 -2i), (2,0), (1,4g+5 -2i), (2,1), ... (1,8g+3), (2,4g - 1 - 2i), (1,Sg + 4): (2,4g - 2i), (3,0)], O 5 i 5 g - 1:

and

[(3,0), (2,49 + 1 - 2i)? (3, l), (2,4g + 2 - 2i), (3,2),. . . (3,49 + 2 + 24, (2,8g + 3), (449 + 3 + 2% (2,8g + 4), (3? 49 + 4 - 2i ) , (2,0), (3,4g + 5 - 24, (2, l), . . . (3,Bg + 3),


(2,4g - 1 - 2 4 , (3,8g + 4) , (2,4g - 2i), ( 1 ,0 ) ] , O 5 i 5 g - 1. An example of the first path is given in Figure 4.4 when g = 1, while examples of the

other two are depicted in Figure 4.5, g = 1, i = 0.

We also repeatedly apply the permutation ( 1 2 3) to the first coordinates of the ver-

tices. Finally, colour the 6g + 3 paths f o d by applying Lemma 1.15 so obtaining 129 + 6

colour-complementary paths. These paths will be called Q-paths. They use al1 the edges

of lengths 29 + 2, 2g + 3, . . . and 69 + 3 in each of the three 2Kagf 5,8g+5.

The proof that this construction is correct is virtually the same as the case when k =

89 + 1, and hence we may conclude that for al1 odd k, 2K3k is decomposable into colour-

complementary paths of length 2k. .

Figure 4.5: Some paths needed to decompw 2Ka9 into cobur-complementary P2&

Lemma 4.2 For odd k, the gmph SK3k+l "S decomposoble into isomorphic copies of any

fied colour-complementory path of length 2k.


Figure 4.6: A path needed to decompose 2K34 into colour-complementary q2.

PROOF. First, note that if k E 1 mod 4, then [E(K3k+i)l is divisible by 2k, and hence

is decomposable into copies of &. Using Lemma 1.15 gives a decomposition of

2K3&+, into colour-complementary paths of length 2k, as required. To finish, we will

consider the case k E 3 mod 4. Let k = 8g+3, where g 2 1. (The case k = 3 was solved completely with Theorem 2.24.)

Label the vertices of 2K3k+1 m, (1, O), (1,1), (1,2), . . . ,(1,8g + 2), (2, O), (2? l ) , (2,2) ,. . . , (2.8g + 2). (3, O), (3, l ) , (3,2),. . . , (3,Bg + 2). Consider the path

[(3.49+2): (1,6g+2), (4,49+3), (1,6g+l),. .. , (3,5g), (1,5g+4), (3,5g+1), (1,59+3):

(3.59+2). (1 ,5gi l ) , (3,5g+3), (1,5g),. . . (3, 6g), (1,4g+3), (3,69+1), (1,49+2), (3,6g+2).

(2.6g + 2). (2: 69 + l), (2,6g + 3), (2, 6g), . . . (2,4g f 3), (2,89 + l) , (2, 4g + 2): (2,8g + 2),

(WI + 1). X, (LO), (1,4g + 11, (w (1,4g), (m, . .- UA - 11, (i,2g + 21, ( ~ 2 9 ~

(1-%+1): (2.2g+l), (3, l ) , (2,2g), (3,2), (2,29-1), ... (3,g-1), (2,9+2), (3,g), (2,g+l),


(3?9 + 21, (2,g), (3,g + 31, (299 - l),. . - (3,291, (2,2)? f3,29 + l) , (2,1)1? as depicted in Figure 4.6 when g = 1.

Obtain colour-complementary paths by repeatedly applying the permutation (1 2 3)

to the first coordinates of the vertices, and (O 1 . . . 89 + 2) to the second coordinates.

We will cal1 these paths P-paths. These patbs use al1 the edges incident with the point oc,

al1 the edges of each of the ~ K Q + ~ , and al1 edges of lengths O, 1, . - . ,2g and 6g + 3,6g + 4,

. . . ,8g + 2 in each of the three 2K89+3,89+3. We also need to consider the 29 + 1 uncoloured paths (as illustrated in Figures 4.7 and

4.8 when g = 1 and i =O):

[(1,0), (2749 + 21, (1, l )? (2,4g + 31, (1, 21, . . . , (2,89)? (1749 - 117 (2789 + 11, (1,49),

(%89 + 21, (L49 + l ) , (3,0), (Z49 + 11, U,89 + 21, (2,491, (1,89 + 11, (2,49 - l), (2,891,

Figure 4.7: A path needed to decompose 2KW into colour-complementary P22.

CHAPTER 4. COLOCIR-COMPLEMENTARY FATHS OF LENGTH 2k, k ODD 31

[(l,o), (2,49-22), (1, l), (2,4g+1-Zi), (1, S),. . . (1,4g+1+2i), (2,8g+I), (174g+2+2i),

(2,89+2), (1,4g+3+23'], (2,0), (1,4g+4+2i), (2,1),. .. (1,8g+l), (2'49-2-21), (1.89+2), (2,49 - 1 - Si), (3, O)], O < i 5 g - 1;

and

[(3?0), (2,4g-2& (3, l), (2,4g+1-2i), (3,2),. . . (3,49+1+2i), (2,8g+l), (3,49+2+29),

(2,89+2), (3,49+3+2i), (2, O), (3,4g+4+2i), (2, l),. . . (3,89+ l ) , (2,4g-2-Si), (3,8g+2),

Figure 4.8: Some paths needed to decompose 2K34 into colour-complementary Pz.

Further, it is necessary to repeatedly apply the permutation (1 2 3) to the first co-

ordinates of the vertices in these 29 + 1 paths to obtain aii the remaining paths needed-

Finaliy, obtain colour-complementary paths by applying Lemma 1.15. These paths will be

caiied Q-paths, and use aü the edges of lengths 29 + 1, 29 + 2, . . . and 69 + 2 in each of the

three copies of 2Ks9+3,8g+3.


Figure 4.9: A path needed to decompose 2Kz2 into colour-complementary Pl+

As might be expected, the permutation (1 2 3) is repeatedly applied to the first c e

ordinates of these vertices, and (O 1 . . . 89 + 6) to the second coordinates to obtain

3(8g + 7) colour-complementary paths known as P-paths. These paths use aii the edges

incident with m, al1 the edges in each of the three copies of 2K8g+7, and all the edges of

lengths O, l9 2, . . . , 29 + 1 and 69 + 6, 69 + 7, . . . and 89 + 6 in each of the t h e copies

2K8g+7,8~+7.

CHAPTER 4. COLOUR-COMPLEMEEITARY PATHS OF LENGTH 26, k ODD 33

We also need to consider the following uncoloured paths (with O 5 i 5 g):

[(&O), (2, lg + 3 - 2i), (1, l ) , (2,4g + 4 - 2i), (1,2),. . . , (1: 49 + 2 + 24, (2,8g + 5 ) ,

(1,4g + 3 + 2i), (2,8g + 6): (L4g + 4 + 2i), (2,0), (1,4g + 5 + 2i), (2, l), . . . , (1,8g + 5),

(2,4g + 1 - 2i), (1,8g + 6), (2,4g + 2 - 2i), (3, O)]

and

[(&O), (%4g + 3 - 2% (3, l) , (2,4g + 4 - 2i), (3,2),. . . (3,4g + 2 + 2t), (2,8g + 5),

(3,4g + 3 + 2i), (2,8g + 6), (3,4g + 4 + 2i), (2,0), (3,4g + 5 + Z i ) , (2, l),. . . (3,8g + 5), (2,4g + 1 - 24, (3,8g + 6), (2,4g + 2 - 2i), (1, O)], as illustrated in Figure 4.10 when g = O

Figure 4. IO: Some paths needed to decompose 2Kz2 into colour-complementary P14.

Again, the permutation (1 2 3) is repeatedly applied to the first coordinates of the

vertices of these paths to obtain 6(9 + 1) paths. These paths must also be "spiit" into colour

complementary paths using Lemma 1.15. These resulting paths wiii be c d e d Q-paths.

These Q-paths use all the edges of length 29 + 2,2g + 3, . . . and 69 + 5 in eafh of the three

copia of 2Kas+r,ss+7. The proof of the validity of these constructions, when k 3 mod 4: is virtualiy identical

the proof in Lemma 4.1. .


Figure 4.11: A path needed to decompose 2KS5 into colour-complementary Pz.

Lemma 4.3 For odd k , the groph 2Kjk as decomposable into isomorphic copies of any fired

wlour-complementay path of length 2k.

PROOF. TO begin, note that there are 5k(5k - 1) edges in the graph 2KSk. If k = 1

mod 4, then the number of edges in the uncoloured complete graph KSk is divisible by 2k.

and hence this complete graph is decomposable into paths of length 2k. As before. this

&es a decomposition of 2Ksk into colour-complementary patbs, by using Lemma 1.15. So,

only the case k = 3 mod 4 need be considered.

First consider k = 89 + 3, with g 2 1. (The case g = O has been handled by Th*

rem 2.24.) Label the vertices of 2K5(e9+3) (1,0), (1, l), (1,2), . . . (1,8g + 2): (2, O), (2, l ) ,

(2,2), . . . (2,8g + 2), . . . , (5, O), (5, l), (5,2), . . . , (5,8g + 2). Then, consider the following

colour-complementary path:

CHUTER 4. COLOIIR-COMPLEMENTARY PATHS OF LENGTH 21c, k ODD 35

Figure 4.12: Some paths needed to decompose 2KS5 into colour-complementary Pz.

each vertex, and the permutation (O 1 . . - 89 + 2) to the second coordinates.

This produces 5(8g + 3) colour-complementary paths but does not use ail the edges of

2KSk. It does use al1 the edges in each of the five copies of 2K89+3, aii of the edges of lengths

0, 1, . . . ,2g, and 6g + 3,69 + 4, . . . , 89 + 2 of five of the copies of 2K8gt3,8g+3: and ail the

edges of length O in the five other copies of 21(89+3,89+3-

We introduce the following uncoloured paths in Kjk, for O 5 z < g - 1, as we11 as their

images after r e p e a t d y applying the permutation (1 2 3 4 5 ) to the first coordinates

of each vertex:


[(l,O), (2,49-2i), (1,1), (2,4g-1-2i), (1,2) ,... (2,8g), (1 ,49+1+2i) , (2 ,8g+l) ,

(1,4g+2+2i), (2,8g+2), (1,4g+3+2i), (2,0), (1,4g+4+2i), (2,l) ,... (2,4g-3-22)?

(1,8g+l), (2,4g-2-2i), (1,8g+2), (2,4g- 1 -2i), (3,0)],

and

[(3, O), (%49 - 2i), (3,1), (2,4g - 1 - Si), (3,% . . (2,8g), (3,49 + 1 + 24, (2,8g + 11, (3, 49 + 2 + 2i), (2,8g + 2), (3,49 + 3 + 2i), (2, O), (3, 49 + 4 + 2i), (2, l), . . . (2,4g - 3 - 24, (3,8g + l) , (2,4g - 2 - 24, (3,8g + 2), (2,4g - 1 - Si), (1,0)]. An example of these two paths

is given in Figure 4.12 when g = 1 and i = 0.

These uncoloured paths produce colour-complementary pat hs by the Lemma 1.15. These

129 paths do not use al1 the edges of 2Ks, but do use ail the edges of lengths 29 + 1,29 + 2,

. . . , 49 and 49 + 2,4g + 3, . . . , 69 + 1 in five of the copies of 2K8,+3,8,+3.

We must also consider edges which have vertices whose first coordinates difier by 2. To

use those remaining, we introduce the following paths, for O 5 i < 2g - 1,

[(1.0), (3,4g -2i), (Al) , (3,49 - 1 - 2i), (1,2) ,... (3?8g), (1,1g+ 1 +2i), (3,8g+ l)?

(1.49+2+2i), (3,89+2), (1,49+3+2i), &O) , (1,49+4+2i), (3,1), ... (3,4g-3-2i),

(1 .8g+l) , (3,4g -2-2i), (1,8g+2), (3,4g- 1 -2i), (5,0)],

and

[(3.0), (1,4g - 2% (3, l ) , (1,4g - 1 - 2i ) , (3,2),. . . (1,8g), (3,49 + 1 + 24, (1,89 + 11,

(3.4g+2+2i), (1,8g+2), (3,4g+3+2i), ( l t O ) , (3,49+4+2z), (1,1), ... (1,4g-3-22),

(3.8g+1). (1,4g-2-2i), (3,89+2), (1,4g-1-22), (4,0)].

Paths of both types are depicted in Figure 4.13, when g = 1 and à = 0.

We also consider the images of these paths after repeatedly applying the permutation

(1 2 3 4 5) to the first coordinates of their vertices, and applying Lernma 1.15 to

obtain colour-complementary pat hs. This uses al1 edges of lengt hs 1, 2,. . . , 49, and 49 + 3,

49 + 4. . . . , 8g + 2 in five of the copies of 2&9+3,8g+3.

But edges still remain. so finaily, we consider the foiiowing paths:

[(q. 0). (5:4g + 21, (4, l), (5, 49 + 31, (4,2),. . . (4: 49 - 11, (5,89 + l), (4,4g)? (5,8g + 21,

(4.49 + 1). (3,0), ('449 + l), (L8g +2), (2,4g), (1,8g + 1): (2,4g - 1):. . . (2,2), (1,4g +3),

(2.1). (1.4g + 2), (2,0)],

and

[(3. O). ( 5 . 9 + 2), (3,l): (5,4g + 3), (3, 2),. . . (3,4g - l), (5? 89 + l), (3,4g), (5,8g + 2):

(3.49 + 1). (LO), (4,4g + 11, (2,8g +2), (4,4g), (2,8g+ l), (4,49 - l),. . . (4,2), (%4g +3),


Figure 4.13: More paths needed to decompose 2KS5 into colour-complementary Pz*.

Figure 4.14: Even more paths needed to decompose 2KJ5 into colour-complementary Pz.

(4, l), (2,4g + 2), (4,O)I: as shown in Figure 4.14 when g = 1.

These paths, and those obtained by repeatedly applying the permutation (1 2 3 4 5)

to the fmt coordinats of their vertices, use al1 the rernaining edges. Ail that remaias is to

form colour-complementary paths fiom them. These p a t b use the only edges kft; namely

those of lengths 49 + 1 and 49 + 2 in each copy of 2Ka,+3,89+3.

Figure 4.15: A path needed to decompose 2K35 into colour-compllementary Pl+

Not surprisingly, the construction for the case k = 89 + 7. with g 1 O, is quite similar.

Fint. label the vertices of 2KH8g+7) (1,O). (1, 1)' (1,2), . . . (1,8g + 6)' (2.0)' (2, 1)' (2,2),

. . . (2,8g + 6), . . . , (5,0), (5, l), (5,2), . . . , (5,89 + 6). Then, consider the following colour-

complementary path (shown in Figure 4.15 when g = 0):

[(4,4g + 3), (2.49 + 3), (3,69 + 4), (2, 49 + 4), (3,6g + 31, (2.49 + 5)' (3,69 + 21, . . . (3,5g + 6). (2,59 + 2), (3, 59 + 5) , (2.59 + 3), ( 3 . 3 + 4), (2,5g t 5), (3'59 + 3). (2.557 + 6).

(3,59+2),... . (2,6g+3), (3.49+5), (2,6g+4), (3,49+4), (2.6g+5), (1,60+5), (1,69+4),

(1.69+6), (1,6g+3), (1,6917) ,... , (1,4g+5), (1,8g+5), (1,4g+4), (1.89+6), (1.4g+3).

(1.0)' ( 1 . b +2), (1, l), (l,49+ l),.. . , (L29 - 1). (1,2g +a). (1, b). (L2g +% (1, 2g + 1).

(2,2g + 11, (%O), (2,29), (3, l), (2,29 - l),.. . , (3,g - 2). (2,g + 21, (3,9 - l), (2.9 + 11, ( 3 d ? ( 2 4 - 11, ( 4 9 + 11, ( 2 4 - 21, (3,g + %. , (3,2g - 21, (2.1). (3:O - 117 (%O),

(3,291, (2,89 + 6): (4,89 + 611.


1

Figure 4.16: Some paths needeà to decompose 2K3j into colour-complementary P14.

Figure 4.17: Some paths needed to decompose 2& into colour-complementary PL4.

As before, we use in our construction colour-complementary paths obtained by repeat-

edly applying the permutation (1 2 3 4 5 ) to the first coordinates of each vertex in

the path and (O 1 -. . 8g + 6) to the second coordinates of each of the vertices of the

resulting paths. These paths use al1 the edges in each copy of 2Kak+r, the edges of lengths

0,1,2, . . . ,2g + 1 and 6g + 6,6g + 7, . . . 8g + 6 in five copies of 2K8g+7,ag+7, and the edges

of length O in the other five copies of 2K8g+7,8g+7.

Next, for O 5 i <_ g, begin with the paths


[(l, O), (2,4g + 3 - 24, (1, l), (2: 49 + 4 - 2i), (1,2),+. . (1: 49 + 2 + 2i), (2,8g + 5),

(1,49 + 3 + 2i), (2,8g + 6): (1,4g + 4 + 2i), (2, O), (1,4g + 5 f Zi), (2, l),. . . (1,8g + 5), (2 ,4g+ l - 2 i ) , (1,8g+6), (2:4g + 2 - 2i}, (3,0)],

and

[(3, O), (2, 4g + 3 - Zi), (3, l), (2,49 + 4 - 2i), (3, S),. . . (3,;lg + 2 + 2i), (2,8g + 5),

(3,4g + 3 + 24, (2,8g + 61, (3,49 + 4 f 3). (2,0), (3,4g + 5 + 2i), (2,1)!. .. (3'89 + 5),

(2,4g + 1 - Si), (3,89 + 6), (2,4g + 2 - 2i), (1, O)], as shown in Figure 4.16 when g = O and à = o.

Also, take the paths resuiting from repeatedly applying the permutation (1 2 3 4 5)

to the fint caordinates in each vertex of these two pathç. This uses al1 the edges of lengths

29 + 2, 29 + 3, . . . ,6g + 5 in five copies of 2 K . ~ + 7 , ~ , + 7 .

Simiiarly, for O 5 i 5 29 consider the paths

[(1,0), (3,4g+2-2i) , (1,1), (3,4g+3-Si) , (1:2) ,... ( 1 7 d g + 3 + 2 i ) , (3,8g+5),

(1,4g + 4 + 2i), (3,8g + 6), (1,4g + 5 + Zi), &O), (1,4g + 6 + 2i), (3, l),.. . (1,8g + 5),

(3'49 - 2i ) , (1,8g + 6), (3,49 + 1 - 24, (5,0)],

and

[(3, O), (1,4g + 2 - Si), (3,1), (1,4g + 3 - 24, (3, S),. . . (3,4g + 3 + Si), (1,8g + 5),

(3,4g + 4 + 2i), (1,8g + 6), (3, dg + 5 + Si), (1,0), (3? 49 + 6 f Zi), (1, l),. . . (3.89 + 5 ) ,

(1,4g - Si), (3,89 + 6), (1,lg + 1 - 2i), (4,O)I. Figure 4.17 gives an example of these two

paths when g = O and i = 0.

Obtain from each of them a further 4 paths by repeatedly applying the permutation

(1 2 3 4 5) to the first coordinates of each vertex. These paths use al1 the edges of

lengths 1, 2, . . . ,4g + 2 and 49 + 5, 4g -t- 6, . . . , 89 + 6 in five copies of 2K8Sir,aS+7- Finally,

to use the edges of Iength 49 + 3 and 49 + 4 in five copies of 2&,+~,~~+7, we need the path

[&O), (5,49+4), (3,l): (5!49+5), (3,2)?... (3,49+1), (5,8g+3), (3,4g+2): (5,8g+6),

(3,4!?+3), (1,0), (4,4g+3), (2,89+6): (4,49+2), (2,8g+% (4,%1+1),. . . (4'21, (2,49+5):

(4,1), (2' 49+4), (4,0)], (illustrated in Figure 4.18 wheng = 0) dong with the paths obtained

by repeatedly applying the permutation ( 1 2 3 4 5) to the first coordinates of each of

its vertices. Finally, for each of the p d h g unc~laured paths, form colour-complementarq. paths

by the application of Lemma 1.15. . Lemma 4.4 For odd k, the gmph 2Ksk+i i s decamposable intci isomotphic copies of any

CIIAPTER 4. COLOUR-COMPLEMENTARY PATHS OF LENGTH 2k, k ODD 41

Figure 4.18: A path needed to decompose 2K3r, into colour-complementary fi4.

fized colour-complementay path of length 2k.


Figure 4.19: A path neeàed to decompose 2Kd6 into colour-compbmentary Pls.

Repeatedly applying the permutation (1 2 3 4 5) to the first coordinates of each

vertex, and the permutation (O 1 . . . 89) to the second coordinates, yields 5(8g + 1)

paths. These paths use al1 the edges incident with the vertex oo, as weU as al1 those

contained in the five copies of 2Kasi1. They aiso use al1 the edges of lengths O? 1, 2, . . . , 29 - 1 and 69 + 2, 6g + 3, . . . , 89 in fiw copies of 2Ksg+l,es+i, and the edges of length O in

the other five copies of 2K8g+L,89+1

This construction does not use al1 the edges of 2K5k+l, so we introduce the foiiowing

pat hs

[(1,0),(2,2g+22+1),(1,1),(2,2g+2i+2),(1,2)~. .. ,(1:6q-22-2): (2,89-1),(1,6g-2P'-

1),(2,~g)7(1169-~~),(2,0),(~,69-2~+1)1(2, 11, ..., (1,8g-1):(2,29+2i-l),(1,89),(2,2g+ 2i~(3,011,0 5 5 g - 1,

and [(2,0),(2,2g+Zi+1),(2, 1),(1,29+2i+2),(2,2), ... ,(2,6g-2i-2), (1,8g-1),(2,69-2i-

111(1, 89),(2,69 -2i),(l , 0),(2,69 -2 i+ l ) ,I i , 11, -- . , - 1),(1, - l),(2,8g),(I, 2g-t

CHAPTER 4. COLOUR-COMPLEMENTARY PATHS OF LENGTH 2k, k ODD

2i),(5,0)], O 5 i 5 g - 1, as illustrated in Figure 4.20 when g = 1 and i = 0.

Figure 4.20: Some paths needed to decompose 2Kd6 into colour-complementary Pie.

These paths are uncoloured, but can be used to produce colour-complementary paths

by Lemma 1.15. We must aiso include the paths produced by repeatedly applying the

permutation (1 2 3 4 5) to the first coordinates of each vertex. This uses al1 the edges

of lengths 29, 2g+ 1, ... 49 - 1 and 49+2, 49+3, ..., 69 + 1 in five of the copies of

2K89+1,89+1. We also use the following path and those that result from the application of

Lemma 1.15 and the afore-mentioned permutation. This gives al1 the edges of lengths 49

and 49 + 1 in five of the copies of 2K89+1,89+1. An example is shown in Figure 4.21 when

g = 1.

[(l.O). (2.49 + l) , (1, l), (2,4g +2), (1,2), . . . (k4g -21, (2,8g - l ) , (1, 49 - 1). (2,8g),

(1.49). (3.0)! (2,4g), (1,89), (2,49 - l ) , (Id39 - 11, (2,4g-2), .-. , (2,2), (1,49+2), (2,1It

(1.4g + 1). (2?0)].

This uses almost al1 the edges of 2KSk+. The final paths required are

[(1.0).(3.2~+2),(1,1),(3,2i+3),(1,2),(3,2i+4),. .. ,(1,8g-2i-3),(3,8g-l),(1,8g-2i- ?).KL89).(1.8g - 2i - 1),(3,0),(1,8g -2i),(3,1), (178g - 2i+l),(3,2) ,... ,(3,2i - 1)!(1,8g - 1).(3.2i).(1.8g)1(3,2i + 1),(5,0)], O 5 i 1 2 9 - 1,

and


Figure 4.21: A path needed to decompose 2Kd6 into colour-complementary Pia.

[(3,0),(1,2i+2),(3,1),(1,21'+3),(3,2),(1,2i+4) ,... , ( 3 ,8g -2 i -3 ) , (1 ,8g -1 ) , (3 ,8g -2 i -

2),(l, 89),(3, 8g - 2i - l),(l,O),(& 8g - 2i)7(1, l) , (3,Bg - 22 + l) ,( l , 21,. . . ,(1,2i - l),(3, 89 - 1),(1,2i),(3, 8g),(l,2i + 1),(4,0)], O 5 i 29 - 1, as shown in Figure 4.22 when g = 1 and

i = O .

We also must include the paths resulting from repeatedly applying the permutation

(1 2 3 1 5) to the first coordinates of each vertex in the path. Finally the paths

obtained produce colour-complementary paths by application of Lemma 1.15. This uses al1

edges of lengths 1, 2, . . . , 8g in five copies of 2K8e1,8g+1.

The second case to consider is k = 8g + 5, with g 2 0. First, we label the vertices of

2Ks(a9+5)+i (1,0), (1,1), (L2)? . . . (L8g +4), (2,O), (2, l), (2,2), .-. (2,8g+4, -. . , (5,O):

(5, 1) , (5,2), . . . , (5,8g + 4), with the remaining vertex being labeled W. We first consider

the case g = O. The graph 2Kz6 can be decomposed into colour-complementary paths of

length 10 as follows. We begin with the colour-complementary path

CHAPTER 4. COLOUR-COMPLEMENTARY PATHS OF LENGTH 2k, k ODD

Figure 4.22: Some paths needed to decompose 2K4s into colour-complernentary PiB.


Figure 4.23: A path needed to decompose 2Ks6 into colour-complementary P26.

This path, and those that result on repeatedly applying the permutation (1 2 3 4 Z)

to the first coordinates of each vertex and (O 1 - . . 89 + 4) to the second coordinates use

many of the edges of 2&(89+83+3. These paths use aii the edges incident with oc, al1 the


edges in each of the five copies of 2K8,+5, and d l the edges of length Or 1, 2, . . . , 29 and

69 + 5,6g + 6, . . . ,8g +4 in five of the copies of 2Keg+5,8g+5, and the eàges of length O in the

other five copies of 2K8g+5,ag+5. We also need the following patbs, dong with those obtained

fiom repeatedly applying (1 2 3 4 5 ) to the first coordinates of the vertices of the paths.

Then, colour the paths according to Lemma 1.15 to achieve colour-complementary paths.

[(1,0),(2,22),(1,1),(2,22),(1,2),. . . ,(1,8g - 22 + 3),(2,8g + 31, ( L 8 q - Si + 4),(2,8g + 4) , ( l , 89-2i+5),(2,0),(1,8g-22+6),(2, l),. . . , (1,89+3),(2, 22-2),(1,8g+4),(2,2i-1),(3, O ) ] ,

g + l < i < 2 g + l ,

[(2,0),(1,2:),(2,1),(1+ 2i),(2,2),. . . ,(2,8g - 2i + 3),(1,8g + 3): (2,Sg - 22' -t 4),( l , 89 + 4),(2,8g-2i+5),(1,0),(2,89-2i+6),(1, l),. . . , (2,8g+3),(1,2i-2),(2,89+4),(1, 22-1),(5,0)],

g + l < i 1 2 g + l

[(1,0),(3,2i),(l,1),(3,22),(1,2) ,..., (1,8g - 22 +3),(3,8g + 3), (1,8g - 2i + 4),(3,8g + 4) , ( l , 89-2i+5),(3,0),(1,89-2i+6),(3, l),. . . , (1,8g+3),(3,22-2),(1,89+4),(3,2i-1),(5, O ) ] ,

1 < i s 2 g + t

and

[(3,0),(1,2i),(3,1),(1,2i),(3,2) ,..., (3,8g - 2 i + 3 ) , ( 1 , 8 9 + 3 ) , (3,8g -2 i+4) , (1 .8g+

4),(3,8g-2~+5),(1,0),(3,89-21'+6),(1, l),. . . , (3,8g+3),(1,2i-2),(3,8g+4),(1,2i-1).(4, O ) ] :

1 5 i 5 29 + 1, al1 of which are depicted in Figure 4.24 with g = 1 and i = 3.

The first pair of paths use d the edges of lengths 29 + 1,29 + 2, . . . ? 69 f 4 in Eve copies

of 2K8g+5,8gi5, while the second pair use al1 the &es of lengths 1, 2, . . . . 8g + 4 im the

other five copies of 2Ka9+5,89+5. 1 With these four lemmq we can now extend the results of Theorems 3.2 and 3.1.

Theorem 4.5 For odd k, 2Kk, r 2 3 can be decomposed into isomorphic copies of any

&ed colour-wmplementary path of length 2k.

PROOF. Note that if r is even, we have exactly the case of Theorem 3.2. So we need

only consider the case when r is odd.

The k t case is r n 1 mod 4. Partition the vertex set of 2Kh into one set of size 5k and

9 of size 4k. Consider the complete ~ b g a p h s induced by these vertices. The graph 2Ksk

can be decornposed into colour-complementary paths of length 2k by Lemma 1.3. The graph

2K4k can be decomposed into colour-complementary paths of Iength 2k by Theorem 3.2. By

Corollary 3.10 we know that K4k,4k and K4k,5k can be decomposed into PQk, and thus to P2k.

CHAPTER 4 . COLOIIR-COMPLEMENTARY PATHS OF LENGTH 2k, k ODD 48

We then have obtained a decomposition of 2KJk,Jk and 2K4k,jk by applying Lemma 1.15 to

these paths.

O

Q P O

Figure 4.24: Some paths needed to decompose 2Ks6 into colour-cornplementary p2&

C H A P T E R 4. COLOUR-COMPLEMENTARY PATHS OF LENGTH 2k, k O D D 49

Next consider r 3 mod 4. Partition the vertices of 2Kkr into one set of size 3k and

9 of size 4k. The complete subgraphs induced by these sets can be decomposed into

colour-complementary paths of length 2k by Lemma 4 . I and Theorem 3.2. The complete

bipartite graphs induced between these sets, 2 K d k , ~ k and 2&k,3k can be decomposed into

colour-complementary paths of length 2k as a result of Corollary 3.10 and Lemma 1.15. 1

Theorem 4.6 For odd k , 2Kkr+,, r 2 2 can be decomposed into isomorphic copies of any

jked colour-complementay path of length 2k.

PROOF. We know this is true when r is even, by Theorem 3.1. If r = 1 mod 4, partition

the vertex set of 2Kk,+~ into one set of size 5k + 1 and 9 of size 4k. By Lemma 4.4, the

complete subgraph on 5k + 1 vertices can be decomposed into colour-complementary paths

of length 2k, and by Theorem 3.2, the complete subgraph on 4k vertices can be decom-

posed into colour-complementary paths of length 2k. The graphs 2K4k+ and 2K4k,5k+l

can be decomposed into cobur-complementary paths of length 2k by Corollary 3.10 and

Lemma 1.15.

The other case, r = 3 mod 4 can be handled similady. Partition the k r + 1 vertices into

one set of size 3k + 1 and 9 of size 4k. The complete subgraphs induced by the former set

can be decomposed into colour-complementary paths of length 2k by Lemma 4.2 and the

latter sets by Theorem 3.2. The graphs K.lk,3k+l and &k,4k can be decomposed into P2k by

Corollary 3.10, giving decompositions of 2K4kJk+1 and 2K4k,4k into colour-complementary

PZk by Lemma 1.15. 1 These two theorems give a very nice corollary.

Corollary 4.7 Let p 6e prime and cr E Z+. Then 2Kn is decomposable into isomorphic

copies of any f ied colour-complernentary path of length 2pa if and only if n > Spa and

n(n - 1 ) r O mod 2pa.

PROOF. If p = 2, we have the case of Corollary 3.3, and are done.

Consider the following: let n = mpQ + r , with p an odd prime, m and a positive integers,

and r an integer such that O 5 r < pQ. Then,


If m is even, then 2pal(m2p20 + (2r - l)mpa). If m is odd. then m2pa and (2r - l )m

are odd, and hence their sum is even and thus, 2pal(m2pa + (2r - l)m)pa. Consequently

for 2pa to divide 1 E(2Kw +,)I, it is necessary that 2p41r(r - 1).

Assuming that r ( r - 1) > O, we see that one of the following must be true. If

2pallE(2Kmp+,)l, either 2palr, 2f l ( r - l ) , 2Ir and pal(r - l), or 2l(r - 1) and palr.

However, each of these is impossible, and thus r ( r - 1) = 0, and r = O or r = 1.

Thus. for 2K, to be divided into colour-cornplernentary paths of length 2pa, by necessity,

either n = mpa or n = mpO + 1. But, since pa is odd, we can apply Theorem 4.5 in the

former case and Theorem 4.6 in the latter.

This result gives further evidence that it will always be possible to decompose 2Kn into

colour-complementary paths of length 2k whenever n(n - 1) E O mod 2k. Further, the

method used to prove Theorems 4.5 and 4.6 will also prove useful in the more general case,

as we see in the next section.

Chapter 5

Lower Bound on Existence of

R e d that for a vertex v of a edge-2-colourd graph H , the degree-vector of v is denoted

r(v) (Definition 1.12), and that a (H) is the least positive integer such that (a(H),a(H))

is an integral linear combination of ~ ( v ) for al1 vertices u in H (Definition 1.13). We again

examine Theorem 1.14.

Corollary 5.1 There ezists a constant no such that 2K, admits a decomposition into iso-

morphic copies of any fized 2-coloured qk for a11 tntegers n > no such that n(n - 1 ) z O

mod 2k.

PROOF. We will apply Theorem 1.14 when H = P2&, where Pzk is a 2-coloured path

with vertices VI, uz, 213, . . . , uzk, V*k+l and k edges of each colour. Examining r(vi) and

T ( v ~ ~ + ~ ) we see that each of these vertices is adjacent to only one other vertex: as such,

their degree-vector is either (1,O) or (O, 1). For any other T ( u ~ ) = (2: O), (0,2), or (1.1).

In particuiar, some vertex must be incident with an edge of each colour: thus for some vj

other than vl or V2k+l, ?(2ij) must be (1: 1). Thus, a(H) = 1. Then. the conditions of

Theorem 1.14 are that n(n - 1) z O mod 2k and n - 1 s O mod 1. The latter condition

is trivial and the former simply says that the number of edges of 2K, is divisible by the

number of edges of PZh-

The foiiowing theorem demonstrates that no need not be particuiarly large when PZk is a colour-complementary path.

CHAPTER 5. LOWER BOUND ON EXISTENCE OF P2k-DECOMPOSITIONç 52

Figure 5.1: A partition of the vertices of 2K20k+m.

Theorern 5.2 Giuen a positive integer k, for al1 n >: Sk, 2Kn can be decomposed cnto

isomorphic copies of any fized colour-complementaty path of lengfh 2k if and only if n(n - 1) = O mod 2k.

PROOF. Assume that n 2 8k. The necessity of the condition n(n - 1) = O mod 2k is

obvious. To verify the condition is suîlicient, let n = 2kr+m, where r 2 4, and O 5 m < 2k.

When rn = O or rn = 1, we obtain the result from Theorems 3.1 and 3.2. Therefore, we

can assume that m 2 2. Since n(n - 1) O mod 2k, we know that c) O mod 2k or

(2) G k rnod 2k. In either case,

CHAPTER 5. LOWER BOUND ON EXISTENCE OF &DECOMPOSITIONS 53

Hence (I) - kr r O! k mod 2k, and thus (7) r O, k mod 2k.

There are naw a number of cases to consider, the fint being r even, and (:) z O mod 2k.

Since r 2 4 is even, we know that 4k12kr. Thus, we shall partition the vertex set of 2K2kr+m

into 5 - 1 sets of size 4k and one set of size 4k + m, as shown in Figure 5.1 when r = 10.

Now, consider

Conseguently, (":m) i O mod 2k. Hence by Th~orem 1.8, we know K4*+,,, bas a

P2k-decomposition, and consequently by Lemma 1.15 2Kdk+, has a colour-complementary

decomposition into paths of length 2k. Also, the graph 2Ksk can be decomposed into

colour-complementary paths of length 2k, by Theorem 3.2.

Thus, we need only concern ourseives with finding a decomposition of 2K4k,4k and

2K4k,4k+m. But this is easily resolved as fmm Coroh"y 3.10, we know there are PZk- decompositions of K2k,4k and K2k,4k+m- We obtain decompositions of 2K2k,4k and 2K. Y . I C + ~

into colour-complementary paths of length 2k by application of Lemma 1.15.

Next, consider the case r even, and (7) r k rnod 2k. In this case, we partition the

vertices of 2K2kr+m into $ - 2 sets of size 415, one of size 6k, and one of size 2k + m, as shown in Figure 5.2 when r = 10. As before, we know how to decompose the graphs 2&,

2Ksk, and 2K2k.4k Ieaving the complete graph 2K2,, and the bipartite graph 2Kssk+,.

Now,

CHAPTER 5. LOWER BOUND ON EXISTENCE OF P?k-DECOMPOSITIONS

Figure 5.2: A partition of the vertices of 2K20k+m,

But (y) - k z O mod 2k, and thus, (":") P O mod 2k, and again we may u ise The

rem 1.8 and Lemma 1.15 to obtain a decomposition of 2Ka+, into coiour-complementary

paths of length 2k. FinallyI by Corollary 3.10 we have a P2k-decomposition for the bipartite

graph Kfigk+rnt and hence a decomposition of 2K2k,2+m hto colour-complementary paths

of length 2k.

CHAPTER 5. LOWER BOUND ON EXISTENCE OF P2k-DECOMPOSITIONS 55

Figure 5.3: Two partitions of the vertices of 2Kirik+,

i fr is odd and (2) i k mod 2k, partition the vertices of 2KZk,+, into Y sets of 4k

vertices and one set of 2k + m vertices, as shown in Figure 5.3 when r = 9. As in the

previous case, (2k:m) E O mod 2k, and by Theorem 1.8 and Lemma 1.15, w are done.

The final case to consider is r odd and (2) r O mod 2k. Partition the set of vertices

into 9 sets of size 4k (here, r 2 5), one of size 6k, and one of size 4k + m, as shown in

Figure 5.3 when T = 9. The decomposition is constructed as above. This compietes the

proof.

Bibliography

[l] B. Alspach and N. Varma, Decompsing complete gmphs into cycles O/ length 2pC,

Annals of Discrete Math 9 (1980) 155-162.

[2] J.C. Bermond, K. Heinrich, and M L . Yu, Ezistence of resolvable path designs, European

J. Combin. 11 (1990), 205-211.

[3] Dame Kathleen Oiierenshaw and Sir Hermann Bondi, The nine prisoners problem, Bull. Inst. Math. Appl. 14 (1978), No. 5-6, 121-143.

[4] H.E. Dudeney, 536 Puzzles and curious problems, Charles Scribner's Sons, New York?

1967.

[5] K. Heinrich, Padh decompositions, Le Matematiche, Vol. XLVII - Fasc. II (1992), 241-

258.

[6] P. Heii and A. Rosa, Graph decompositions, handcufed designs, and balanced P-designs,

Discrete Math. 2 (1972), 229-252.

[7] J.D. Horton, Resolvable path designs, J. Combinatorial Theory A 39 (1985), 117-131.

[8] C. Huang and A.Rosa, Decomposition of complete gmphs into tees , Arc Combinatoria

5 (1978): 23-63.

[9] E.R. Larnken and RM. Wilson, Decompositions of edge-coloured complete graphs, J . Combin. Theory Ser. A 89 (2000), 144-200.

[IO] C. Parker, Complete bipartite gmph path decompositions, PhD Thesis, -4uburn Univer-

sity, (1998).

CHAPTER 5. LOWER BOUND ON EXISTENCE OF" P2it-DECOMPOSITIONS 57

[Il] M. Tarsi, Decomposition of a complete multigmph into simple paths: Nonhlanced hand-

cuffed Designs, J. Combin. Theory Ser. A 34 (1983), 60-70.

[12] M. Tniszczynski, Decomposition of XK,,, into paths, Discrete Math 55 (1985), 89-96.

[13] M.L. Yu, On pdh factorizutions of complete multipartite gmphs, Discrete Math. 122

(19931, 325-333.

Documents

Two-coloured Path Decompositions