Achromatic number of fragmentable graphs

Achromatic Number ofFragmentable Graphs

Keith Edwards

SCHOOL OF COMPUTINGUNIVERSITY OF DUNDEE

DUNDEE DD1 4HN, UNITED KINGDOME-mail: [email protected]

Received June 2, 2008; Revised August 17, 2009

Published online 6 December 2009 in Wiley Online Library (wileyonlinelibrary.com).DOI 10.1002/jgt.20468

Abstract: A complete coloring of a simple graph G is a proper vertexcoloring such that each pair of colors appears together on at least oneedge. The achromatic number �(G) is the greatest number of colors insuch a coloring. We say a class of graphs is fragmentable if for any positive�, there is a constant C such that any graph in the class can be brokeninto pieces of size at most C by removing a proportion at most � of thevertices. Examples include planar graphs and grids of fixed dimension.Determining the achromatic number of a graph is NP-complete in general,even for trees, and the achromatic number is known precisely for only veryrestricted classes of graphs. We extend these classes very considerably,by giving, for graphs in any class which is fragmentable, triangle-free, andof bounded degree, a necessary and sufficient condition for a sufficientlylarge graph to have a complete coloring with a given number of colors. Forthe same classes, this gives a tight lower bound for the achromatic numberof sufficiently large graphs, and shows that the achromatic number can bedetermined in polynomial time. As examples, we give exact values of theachromatic number for several graph families. � 2009 Wiley Periodicals, Inc. J Graph

Theory 65: 94–114, 2010

Keywords: achromatic number; detachment; fragmentable graph

Journal of Graph Theory� 2009 Wiley Periodicals, Inc.

94

ACHROMATIC NUMBER OF FRAGMENTABLE GRAPHS 95

1. INTRODUCTION

A complete coloring of a simple graph G is a proper vertex coloring such that eachpair of colors appears together on at least one edge. The achromatic number �(G) isthe greatest number of colors in such a coloring.

These concepts were introduced by Harary et al. [10]. There are two surveys of theachromatic number [6, 11]; the first also covers the harmonious chromatic number.

We say a class of graphs is fragmentable [8] if for any positive �, there is a constantC such that any graph in the class can be broken into pieces of size at most C byremoving a proportion at most � of the vertices.

As shown in [8], any class of graphs with a suitable separator theorem is frag-mentable. This includes any class of graphs of bounded genus, indeed any class witha fixed excluded minor [1]. Also fragmentable are classes such as lattice graphs ofbounded dimension, such as square and cubic lattices.

Although the achromatic number has a long history, there are very few classes ofgraphs for which its value is known precisely; namely bounded degree trees [4], pathsand cycles, and collections of paths or cycles [16, 17]. On the other hand, it wasshown to be NP-complete for general graphs by Farber et al. [9], for cographs andinterval graphs by Bodlaender [2], and finally even for trees [3]. Other recent work hasconcentrated on algorithms for approximating the achromatic number [5, 13–15]. Ithas also been shown [12] that under a suitable complexity assumption, the achromaticnumber cannot, in general, be approximated by a factor of better than

√ln(n).

In this article we use results from an earlier article [7] to extend very considerablythe classes of graphs for which a precise value of the achromatic number can bedetermined, at least for sufficiently large graphs. We show that the achromatic numbercan be determined in polynomial time for graphs in any class which is fragmentable,triangle-free, and of bounded degree. We do this by giving a necessary and sufficientcondition, checkable in polynomial time, for a graph in such a class to have a completecoloring with a given number of colors. This condition unifies and extends the resultsmentioned above. It also allows us to give tight lower bounds on the achromatic numberfor the same classes of graphs, and, as examples, to compute the exact value for somefamilies of graphs.

2. NOTATION AND TERMINOLOGY

In this section we will give formal definitions of the terms complete coloring, achro-matic number, detachment and exact coloring, and introduce some further notation andterminology. We denote by E(X) the set of edges of a graph with both endpoints in X,and by E(X,Y) the set of edges joining a vertex of X to a vertex of Y .

A. Complete Coloring and Achromatic Number

A complete coloring is a function f from a color set C to the set V(G) of verticesof G such that for any edge e of G, with endpoints x, y say, f (x) �= f (y), and for each

Journal of Graph Theory DOI 10.1002/jgt

96 JOURNAL OF GRAPH THEORY

pair of distinct colors c, c′, there is at least one edge (x,y) such that x has color cand y has color c′. The achromatic number �(G) is the greatest number of colors in acomplete coloring of G.

Remark. Note that the (usual) chromatic number is the least number of colors ina complete coloring.

Definition of q(m) and r(m). Let m be a positive integer. Then we define q(m) tobe the greatest integer k such that

(k2

)≤m. We also define r(m) to be m−(q(m)2

).

Remark. It is easily calculated that

q(m)=⌊

1+√8m+1

2

⌋.

Also 0≤r(m)<q(m).

By comparing the number of edges of G with the number of color pairs, it followsimmediately that for any graph G with m edges, �(G)≤q(m).

B. Harmonious Coloring

A harmonious coloring is a function f from a color set C to the set V(G) such thatfor any edge e of G, with endpoints x, y say, f (x) �= f (y), and for each pair of colorsc, c′, there is at most one edge (x,y) such that x has color c and y has color c′. Theharmonious chromatic number h(G) is the least number of colors in a harmoniouscoloring of G.

Similar to q(m),r(m) above, it is useful to define Q(m) to be the least positive integerk such that

(k2

)≥m, and R(m) to be(Q(m)

2

)−m. Then

Q(m)=⌈

1+√8m+1

2

⌉

and it is easily shown that h(G)≥Q(m) for any graph with m edges.

C. Exact Coloring

An exact coloring of G with n colors is a proper vertex coloring of G with n colors inwhich for each pair of colors c, c′, there is exactly one edge (x,y) such that x has colorc and y has color c′.

In all three cases, if the coloring concerned uses k colors, we will refer to a complete(resp. harmonious, exact) k-coloring.



D. Detachment

Let G be a simple graph with vertex set v1, . . . ,vn. Then a graph H is a detachment ofG if the vertex set V(H) can be partitioned into n sets V1, . . . ,Vn such that (i) for eachi, Vi is an independent set, and (ii) for each i, j, i �= j,

|E(Vi,Vj)|={

1 if (vi,vj)∈E(G)

0 otherwise.

In the case that G is the complete graph Kn, condition (ii) simply becomes|E(Vi,Vj)|=1 for each i, j, i �= j.

Remarks.

(1) A simple graph H has an exact n-coloring if and only if it is a detachment ofKn; the color classes correspond to the vertex sets V1, . . . ,Vn.

(2) An exact coloring is one which is both harmonious and complete. If G has medges and has an exact coloring, then m=(n2) for some n, and �(G)=h(G)=q(m)=Q(m)=n.

E. Relationship Between Colorings

The relationship between exact coloring and harmonious coloring is simple. If G is agraph with at most

(k2

)edges, define G(k) to be the graph G∪(

(k2

)−|E(G)|)K2, i.e. G

with sufficient isolated edges added to ensure that G(k) has exactly(k

2

)edges. Then G

has a harmonious k-coloring if and only if G(k) has an exact coloring (which necessarilyalso uses k colors). The harmonious chromatic number h(G) is the smallest k for whichG(k) has an exact coloring.

The relationship between exact coloring and complete coloring is, however, morecomplicated. If G has at least

(k2

)edges, then G has a complete k-coloring if and only

if there is subgraph G′ of G with(k

2

)edges, such that G′ has an exact k-coloring in

which for each edge e∈E(G)−E(G′), the endpoints of e have distinct colors.It is not sufficient merely that there be a subgraph with an exact k-coloring; for

example the path P4 has an exact 3-coloring (e.g. 3,1,2,3), but the cycle C4 does nothave a complete 3-coloring and �(C4)=2.

The number of possible subgraphs of G with(k

2

)edges is in general exponential in m.

Thus even for classes for which the existence of an exact coloring can be determinedin polynomial time, we cannot obtain a polynomial time algorithm for the achromaticnumber simply by testing each possible subgraph. Fortunately, however, for the classesof graphs considered in this article, it is not necessary to do this, and so, as will beshown in Section 4, a polynomial algorithm can be obtained.

F. Partial Harmonious Coloring

A partial harmonious coloring of a graph G is a coloring of a subset of the vertices ofG which is a harmonious coloring of the graph induced by the colored vertices, and issuch that no uncolored vertex has two or more colored neighbors with the same color.



We will say that a pair s, s′ of colors occurs on or is used on an edge e if theendpoints of e have colors s and s′.

We say that an edge is colored if both of its endpoints are colored, otherwise wesay it is uncolored.

G. Degree Sum and Degree Excess

Let S be a subset of the vertices of a graph G. Then we define the degree sum of S,denoted as ds(S,G), to be

∑v∈S dG(v), where dG(v) denotes the degree of v in G.

If V(G) is partitioned into r parts V1, . . . ,Vr, the degree excess of Vi, dexG(Vi), isgiven by dexG(Vi)=ds(Vi,G)−(r−1).

H. Fragmentable Graphs

Informally, a graph is fragmentable if the removal of a small proportion of the verticescan break it up into bounded sized pieces.

Formally, we will say that a graph G= (V ,E) is (C,�)-fragmentable if there is a setX ⊆V such that (i) |X|≤� |V| and (ii) every component of G\X has at most C vertices.

We say that a class � of graphs is fragmentable if for any �>0, there is a constantC� such that every graph in � is (C�,�)-fragmentable.

I. Exact Graph

It is frequently useful to consider graphs which have exactly(r

2

)edges for some

integer r. We shall call such a graph as an exact graph.

3. BOUNDED DEGREE GRAPHS

In this section we give, for certain bounded degree classes of graphs, some necessaryand sufficient conditions for the existence of a complete coloring with a specifiednumber of colors. We will assume throughout that there are no isolated vertices, asthese make no difference to the existence of a complete coloring.

To do this, we need to quote some terminology and results from [7] concerning theexistence of an exact coloring of such graphs.

A. Rare Vertex Degrees

Let G be a graph with maximum degree d, and i be a positive integer. Let ni be thenumber of vertices of degree i in G, i.e. ni =|{v :d(v)= i}|. Let Md = lcm{1, . . . ,d}, andlet Rd = (d3 +d2)M2

d +(d3 +d2 +2d)Md. In [7], we define a rare vertex degree to bea degree i such that 0<ni ≤Rd. We also define RG to be the set of vertices whichhave rare degree.



Remark. If G is small it may be that every vertex has rare degree; however, weshall be interested only in graphs which are large enough to ensure that some degreeis not rare. Note that |RG|≤dRd.

B. Degree Partition Extension Condition

Let G be a graph with(r

2

)edges and maximum degree d. Then G satisfies the degree

partition extension condition if and only if there is a partition � of the vertices into rsets V1, . . . ,Vr such that for each i=1, . . . ,r, the sum of the degrees of the vertices inVi is r−1, and such that the function c� :RG →{1, . . . ,r} given by

c�(v)= i⇐⇒v∈Vi

is a partial harmonious coloring of G.We will need the following theorem [7].

Theorem 3.1. Let d be a positive integer, and �1>0 be a real number. There exists areal number �0>0 such that for any positive integer C, there is an integer N for whichthe following holds: if G= (V ,E) is a graph with maximum degree d, satisfying thefollowing properties:

1. G is an exact graph with(r

2

)edges, where r≥N;

2. G is (C,�0)-fragmentable;3. At least �1 |V| vertices of G have an independent neighborhood set;

then G is a detachment of Kr if and only if G satisfies the degree partition extensioncondition.

For fragmentable classes of graphs, we have the following corollary.

Corollary 3.2. Let � be a fragmentable class of triangle free graphs, and d be apositive integer. Then there is an integer N, such that if G∈� is an exact graph with(r

2

)edges and maximum degree at most d, and r≥N, then G is a detachment of Kr if

and only if G satisfies the degree partition extension condition.

As noted in [8], any class of graphs with a suitable separator theorem is fragmentable.These include any class of graphs of bounded genus, indeed any class with a fixedexcluded minor [1]. Also fragmentable are classes such as lattice graphs of boundeddimension, e.g. square and cubic lattices.

C. Definition of Scarce Degree

Recall that a vertex degree i is rare if the number ni of vertices of degree i satisfies0<ni ≤Rd, where d is the maximum degree. For complete coloring, we need a slightlydifferent condition, so we will say that a vertex degree i≥1 is scarce if the number ni

of vertices of degree i satisfies 0<ni ≤Rd(d+1), where d is the maximum degree. LetSG denote the set of vertices whose degrees are scarce.



D. Definition of Pd

We now use Lemma A.1 to define a constant Pd depending only on d. Define twofunctions f1, f2 by

f1(x,y) = (2d+1)(x2+2d max(x,Rd(d+1))+(d+1)y),

f2(x,y) = Rd(d+1)+1+(d+1)(x2+2d max(x,Rd(d+1))+(d+1)y).

Let F be the function given by Lemma A.1, and let N1 = (d+1)(2d −1) and N2 =d.Then set Pd = ((d+1) / (2d+1))f1(F(N1,N2),F(N1,N2)).

We now give a condition, similar to the degree partition extension condition, whichwill be used to determine whether or not a graph has a complete, rather than exactcoloring.

E. Partition Condition With r Parts

Let G be a graph with m edges and maximum degree d. Suppose that r≥q(m)−d. ThenG satisfies the partition condition with r parts if and only if there is a set of verticesP⊆V with |P|≤Pd, a set of edges Edel of the graph induced by P, and a partition �of the vertices of the graph G′ =G−Edel into r sets V1, . . . ,Vr satisfying the following:

(P1) for each i, 1≤ i≤d, the number of vertices of degree i in V \P is either zeroor is greater than Rd(d+1);

(P2) for each i=1, . . . ,r, the sum of the degrees (in G′) of the vertices in Vi, ds(Vi,G′),is at least r−1;

(P3) the function c� :P→{1, . . . ,r} given by


is a partial harmonious coloring of G′ such that for any edge e= (u,w)∈Edel,c�(u) �=c�(w) (thus c� is a partial proper vertex coloring of G, not just of G′);

(P4) the degree excesses, dexG′(Vi)=ds(Vi,G′)−(r−1) for each i=1, . . . ,r, satisfydexG′(Vi)≤

∑j�=i dexG′(Vj) for each i.

Remark. Note that condition (P1) ensures that SG ⊆P and SG′ ⊆P.

Theorem 3.3. Let d be a positive integer, and �1 ∈ (0,1) be a real number. Thereexists a real number �0>0 such that for any positive integer C, there is an integer M forwhich the following holds: for any graph G= (V ,E) with maximum degree d satisfyingthe following properties:

(1) G has m≥M edges;(2) G is (C,�0)-fragmentable;(3) At least �1 |V| vertices of G have an independent neighborhood set;

then for r≥q(m)−d, G has a complete coloring with r colors if and only if G satisfiesthe partition condition with r parts.



Proof. Fix an integer d and an �1>0. The result is obvious if d=1; so we assumethat d≥2. Then apply Theorem 3.1 to d(d+1) and �1 /2 to obtain the real number2�0>0. Let C be a positive integer, and suppose that G is a graph satisfying

1. G is (C,�0)-fragmentable;2. At least �1 |V| vertices of G have an independent neighborhood set.

Let r≥q(m)−d.Suppose first that G satisfies the partition condition with r parts, and let P, Edel,

� be, respectively, a vertex set, edge set, and vertex partition satisfying conditions(P1)–(P4). We will show that G has a complete r-coloring. First note that the total degreeexcess of the parts of � is 2|E(G′)|−r(r−1)≤2m−r(r−1), and since r≥q(m)−d,we have

2m−r(r−1) = 2

(m−

(r

2

))

= 2

((q(m)

2

)+r(m)−

(r

2

))

≤ 2q(m)+2

((q(m)

2

)−(

r

2

))

≤ 2(r+d)+2

((r+d

2

)−(

r

2

))

= 2(r+d)+2

(rd+

(d

2

))= (2d+2)r+d(d+1).

Provided that m is large enough, each part of � contains vertices not in P, and so wecan repeatedly move a vertex from the part with high degree sum to the part with lowestdegree sum if necessary to ensure that the difference of the degree sums is at most d(this will not affect conditions (P1)–(P4)). Hence, if m, and so r, is large enough, wecan assume that the largest degree excess is at most 4d.

We now wish to remove certain edges from G′ and identify some vertices, to givean exact graph G∗ with exactly

(r2

)edges.

Note that by condition (P4) and the fact that the total degree excess is even, thedegree excesses of the parts of the partition form the degrees of a loopless multigraphM(�) with the parts as vertices, and maximum degree at most 4d.

Now as above let Md = lcm{1, . . . ,d}. An i-block will be a set of Md / i vertices, allof degree i, which all lie in the same part of the partition. We will refer to a block tomean an i-block for some i.

Now select a set of O(n) edges which are at a distance at least 3 apart from eachother, and from any vertex in P. Since we have O(n) such edges and there are only ddifferent vertex degrees, we can choose a subset I of O(n) edges such that all of theedges in I have endpoints with degrees d0, d1 say (where d0, d1 may be equal).



Provided m is large enough, we can, by swapping blocks (not containing any vertexof P) between parts of the partition if necessary, assume that each part of � containsO(r) vertices of each of the degrees d0 and d1. Then again by swapping vertices of thesame degree between parts, we can ensure that, for each edge ijt (the tth edge joiningi and j) of the multigraph M(�) above, there is an edge eijt =uw∈ I with u∈Vi andw∈Vj. Denote by E� the set of all such edges eijt.

Now for each edge uw∈E�∪Edel, with u∈Vi and w∈Vj, we can pick a pair of edgese1(uw)=v0v1,e2(uw)=v2v3 ∈ I, and by swapping vertices of the same degree betweenparts, we can ensure that v1 ∈Vi, v2 ∈Vj, and v0,v3 ∈Vk for some k �= i, j. Consider theset S of all the edges e1(uw), e2(uw) together with the edges which have both ends in P.Since, for each i, the number of edges of E�∪Edel with an endpoint in Vi is bounded,we can choose the values of k so that no two edges in S have endpoints in the samepair Vx,Vy of parts of the partition. Now identify vertices u and v1, w and v2, v0 andv3 (the combined vertices remain in Vi, Vj and Vk, respectively), and if uw∈E�, deletethe edge uw from the graph G′. Note that if two vertices formed by these identificationsare in the same set Vi, they cannot have a common neighbor.

This forms the graph G∗, with a partition �∗. Some vertices may now have degreegreater than d, but at most d(d+1).

Now G∗ has exactly(r

2

)edges, and maximum degree d∗ with d≤d∗ ≤d(d+1). We

claim that every vertex of rare degree in G∗ is one of the following: (i) an originalvertex in P, or (ii) the result of an identification. For if vertex v does not satisfy (i)or (ii), then v is not the result of identifications, and is not in P. Hence v∈V \P, soeither v has degree d0 or d1, or the number of vertices of degree d(v) in V \P has notchanged. However, neither of the degrees d0,d1 (which may not be distinct) can berare in G∗, since there were O(n) edges in I of which only O(q(m))=O(

√n) have been

changed. Hence in either case d(v) is not a rare degree in G∗.It follows that the partition �∗ gives a partial harmonious coloring of the rare degree

vertices of G∗.It is also clear that only O(

√m) vertices are affected by the deletions and identifica-

tions used to obtain G∗ from G. Hence it follows easily that G∗ is (C,2�0)-fragmentable,and that at least (�1 /2) |V(G∗)| vertices of G∗ have an independent neighborhood set.

It follows from Theorem 3.1 that provided m, and therefore r, is large enough, G∗ hasan exact coloring with r colors. Undoing the identifications and replacing the deletededges then immediately give a complete coloring of G with r colors, as required.

Conversely, suppose that G has a complete coloring c0 with r colors. Define thepartition �0 by letting Wi be the set of vertices with color i, for each i=1, . . . ,r, andfor any vertex v, denote by c0(v) the color class Wi containing v. There are two cases.

Case (1). Suppose that there is a set S1 consisting of |SG| color classes with totaldegree excess at least 2d|SG|. Let S2 be the set of all color classes containing somevertex of scarce degree (note that |S2|≤|SG|). Let S=S1 ∪S2. (Note that SG may beempty, in which case S1, S2, S are all empty as well.) We will redistribute the verticesin these color classes, as follows: first put each scarce degree vertex into a distinctcolor class. Then go through the rest of the vertices in turn, and assign each to thecolor class with the lowest (current) degree sum. Then when all the vertices have been



reassigned, the degree sums cannot differ by more than d. Hence if the least degreeexcess among these color classes is now �, then the total degree excess of these colorclasses is at most (�+d)|S|≤2(�+d)|SG|. But by assumption, the total degree excess isat least 2d|SG|, so we must have �≥0, so that every color class has degree sum at leastr−1. Now if any color classes Wi,Wj (not just those in S) have degree sums satisfyingds(Wi,G)>ds(Wj,G)+d, then move some non-scarce-degree vertex from color Wi toWj. We can do this repeatedly until all degree sums differ by at most d. Let � bethe resulting partition, and take P=SG, Edel =∅. Then it is easy to see that conditions(P1)–(P3) are satisfied. To see that (P4) is satisfied, note that if SG is empty, then thepartition is unchanged, apart from moving some vertices to a color class with lowerdegree sum, so (P4) must be satisfied. Otherwise, |SG|≥1, so the total degree excessis at least 2d. If the least degree excess in � is �, then largest degree excess is at mostd+�. If (P4) is not satisfied, we must have d+�>(r−1)�, which, provided r is largeenough, forces �=0. But then the largest degree excess is at most d, which is at mostthe sum of the degree excesses of the other classes.

Case (2). Now suppose that every set of |SG| color classes has total degree excessless than 2d|SG|. Let X be a set of edges of G which use every color pair exactly once,and let Y =E(G)−X.

We first partition the set of colors classes as follows: For each non-empty subsetI ⊆{1, . . . ,d}, and each j∈{0, . . . ,d}, let SI,j be the set of color classes Wk such that thedegree excess dexG(Wk) is equal to j (or, in the case that j=d, greater than or equal to j),and such that the set of vertex degrees occurring in Wk equals I, i.e. {d(v)|v∈Wk}= I.

Now enumerate the pairs (I, j) as p1,p2, . . . so that |Sp1 |≤|Sp2 |≤· · ·, and let ai=|Spi |.Also let V (i) be the set of vertices of degree i, and let r1, . . . ,rd be a permutation of1, . . . ,d such that |V (r1)|≤· · ·≤|V (rd)|. Let bi =|V (ri)|. We wish to find indices L,M,such that the sets SpL , V (rM) are, in a suitable sense, much bigger than all the precedingones. As above, define two functions f1, f2 by

f1(x,y) = (2d+1)(x2+2d max(x,Rd(d+1))+(d+1)y),

f2(x,y) = Rd(d+1) +1+(d+1)(x2+2d max(x,Rd(d+1))+(d+1)y)

and let N1 = (d+1)(2d −1) and N2 =d. Let F be the function given by Lemma A.1.Then by Lemma A.1 either (a) there exist L≤N1, M ≤N2, such that (i)

aL ≥ f1

(∑j<L

aj,∑

j<Mbj

),

bM ≥ f2

(∑j<L

aj,∑

j<Mbj

)

and (ii) ∑j<L

aj ≤F(L−1,M−1)≤F(N1,N2),



or (b) we have aN1 ≤F(N1,N2) or bN1 ≤F(N1,N2). In the latter case (b), either thenumber of vertices or the number of colors is bounded by a constant depending on d;hence, the number of vertices is bounded. Thus if the graph is large enough, case (a)must hold.

Let C=⋃j<L Spj so that |C|=∑j<L aj ≤F(N1,N2), and let V∗ =⋃j<M V (rj), so that|V∗|=∑j<M bj ≤F(N1,N2).

Let VC be the set of vertices colored with a color in C. Let S′′ =VC ∪V∗, and let E′be the set of edges which have both endpoints in S′′. Let S′ be the set of endpoints ofthe edges in E′. Finally let S=V∗∪S′ and let P=S∪N(S).

Now if |C|≤|SG|, then the total degree excess of the color classes in C is at most2d|SG|≤2dRd(d+1); otherwise, we could add extra classes to C so that it contains |SG|classes and use case (1). On the other hand, if |C|>|SG|, then the total degree excess isat most 2d|C|, since otherwise we could choose a subset of |SG| color classes with totaldegree excess greater than 2d|SG| and again use case (1). Hence, the total degree excessof the color classes in C is at most 2d max(|C|,Rd(d+1)). Hence since c0 is a completecoloring, the number of edges with both ends in VC is at most

(|C|2

)+d max(|C|,Rd(d+1)).Any vertex in S is either an endpoint of one of these edges or is in V∗∪N(V∗). Hence

|S| ≤ 2

[(|C|2

)+d max(|C|,Rd(d+1))

]+(d+1)|V∗|

≤ |C|2 +2d max(|C|,Rd(d+1))+(d+1)|V∗|

= 1

2d+1f1(|C|, |V∗|)

≤ 1

2d+1f1(F(N1,N2),F(N1,N2))

= 1

d+1Pd.

So |P|≤ (d+1)|S|≤Pd.Now we have

|SpL |=aL ≥ f1

(∑j<L

aj,∑

j<Mbj

)= f1(|C|, |V∗|)≥ (2d+1)|S|≥|P|+|N(S)|,

|V (rM)|=bM ≥ f2

(∑j<L

aj,∑

j<Mbj

)= f2(|C|, |V∗|)>Rd(d+1) +|P|.

Recall that X is a set of edges of G which use every color pair exactly once, andthat Y =E(G)−X. Let Edel be the set of edges in Y which are incident with a vertexin S, and let G′ =G−Edel.

Now if v∈N(S)\S, the color c0(v) of v is not in C (for then, by the definition of S, vwould be in S). Hence c0(v)∈Sp�

for some �≥L. Let p� = (I, j), so that dG(v)∈ I, and



c0(v) has degree excess j. |Sp�|≥|SpL |≥|P|+|N(S)|. Every color in Sp�

has degreeexcess j (or at least j if j=d) and contains a vertex of degree d(v).

Thus if N(S)\S={v1, . . . ,vz}, we can find distinct color classes W1, . . . ,Wz andvertices u1, . . . ,uz such that (i) ui ∈Wi, (ii) dG(ui)=dG(vi) for each i, (iii) dexG(Wi)=dexG(c0(v)) or both are at least d, and (iv) none of the color classes W1, . . . ,Wz containsa vertex of P.

Now, for each i=1, . . . ,z, swap the colors of vi and ui. Denote by V ′i the set of

vertices now colored i. (This coloring, which may not be complete or even proper, willsatisfy (P1)–(P3) but possibly not (P4).)

Finally, as above, if any color classes V ′i , V ′

j have degree sums satisfyingds(V ′

i ,G′)>ds(V ′

j ,G′)+d, then move some vertex not in P from color class V ′

i to V ′j .

We can do this repeatedly until all degree sums differ by at most d.Denote by � the partition of the vertices induced by this final coloring, and let Vi

be the set of vertices colored i. We now need to show that P, Edel, and � satisfy theconditions (P1)–(P4).

(P1) Suppose that there is a vertex v of degree i in V \P. Then v �∈V∗, so V (i) =V (rj)

for some j≥M. Thus |V (i)|≥|V (rM)|>Rd(d+1)+|P|, so the number of vertices of degreei in V \P is greater than Rd(d+1), as required.

(P2) For any vertex v, let del(v) be the number of edges in Edel incident with v,so that dG′(v)=dG(v)−del(v). Note that for any v, dexG(c0(v))≥del(v) since c0 is acomplete coloring of G.

We first establish that ds(V ′i ,G

′)≥r−1 for each i. There are three cases to consider:Case (i): Wi contains some vertex of P. Then we cannot have uj ∈Wi for any j, since

the uj are chosen to be in parts of the partition �0 which do not contain elements ofP. Hence V ′

i is obtained from Wi by replacing a (possibly empty) set of the verticesvj by the corresponding uj. Since del(uj)=0, it is clear that ds(V ′

i ,G′)≥ds(Wi,G), and

ds(Wi,G)≥r−1 since Wi is a color class of a complete coloring of G.Case (ii): Wi contains a vertex uj. Then V ′

i =Wi \{uj}∪{vj}. Then either dexG(Wi)≥d≥del(vj), or

del(vj)≤dexG(c0(vj))=dexG(c0(uj))=dexG(Wi)

so

ds(V ′i ,G

′)=ds(Wi,G)−del(vj)≥ds(Wi,G)−dexG(Wi)=r−1.

Case (iii): Wi does not contain any element of P or any vertex uj. Then V ′i =Wi and

all the degrees of these vertices are unchanged so ds(V ′i ,G

′)=ds(Wi,G)≥r−1.The final recoloring only makes the degree sums more equal, so it follows that

ds(Vi,G′)≥r−1 for each i.(P3) The function c� :P→{1, . . . ,r} given by


defines a partial coloring of G′. Note that for each v∈S, c�(v)=c0(v). Now since c0

is a complete coloring of G, and we have deleted the set Edel of edges, and recolored



the vertices of N(S)\S with distinct colors not used on S, it follows that c� is apartial harmonious coloring of P such that for any edge (u,w)∈Edel, c�(u) �=c�(w), asrequired.

(P4) If no color not in C has positive degree excess, then Edel will be the whole ofY , and G′ will be an exact graph, with all color classes Vi having degree excess 0.

Otherwise, some set Sp�, �≥L, contains colors of positive degree excess; hence, the

total degree excess is at least Rd(d+1)>2d. Then as before, since the degree excessesdiffer by at most d, the largest is at most the sum of the rest, as required. �

We obtain the following corollary:

Corollary 3.4. Let � be a fragmentable class of triangle free graphs, and d a positiveinteger. Then there is an integer M, such that if G∈� is a graph with m≥M edges,and maximum degree at most d and r≥q(m)−d, then G has a complete coloring withr colors if and only if G satisfies the partition condition with r parts.

It is important to note that, for graphs of bounded degree, the partition conditionwith r parts can be tested in polynomial time:

Lemma 3.5. Let d be a fixed positive integer. Let G be a graph with m edges andmaximum degree d, and suppose that r≥q(m)−d. Then it can be tested in polynomialtime whether or not G satisfies the partition condition with r parts.

Proof. The graph G satisfies the partition condition with r parts if and only if thereis a set P⊆V(G) with |P|≤Pd, a set Edel ⊆E(P), and a partition � of V(G−Edel) intosets V1, . . . ,Vr, satisfying conditions (P1)–(P4).

Consider a triple (P,Edel,�) satisfying all these conditions, and let T =|P|, andset U =V \P. Then we may assume (by renaming the parts of the partition if neces-sary) that every vertex in P is in one of V1, . . . ,VT . Also note, as in the proof ofTheorem 3.3, that the total degree excess is at most (2d+2)r+d(d+1), and that,without affecting conditions (P1)–(P4), we can if necessary repeatedly move a vertexnot in P from the part with the highest degree sum to the part with the lowest degreesum to ensure that the difference of the degree sums is at most d, and the largestdegree excess is at most 4d. Also by reordering the parts if necessary, we can assumethat dexG(VT+1)≥dexG(VT+2)≥·· ·≥dexG(Vr). It follows that the number of possiblevectors (dexG(V1), . . . ,dexG(Vr)) of degree excesses is bounded by a polynomial in r.

Since |P| is bounded, the number of possible sets P satisfying (P1) is polynomial.For a given P, the number of sets Edel is bounded, and the number of partial harmoniouscolorings c� (or, equivalently, the number of possible restrictions �|P of the partition� to the set P) is also bounded. Hence the number of possible quadruples, consistingof P, Edel, �|P, and the vector (dexG(V1), . . . ,dexG(Vr)), which we need to consider,is bounded by a polynomial in r. Now G satisfies the condition if and only if for atleast one such quadruple, we can partition the vertices of U into r sets U1, . . . ,Ur suchthat for each Uk, k=1, . . . ,r, the sum

∑v∈Uk

d(v) is Bk, where Bk =r−1+dexG(Vk)−∑v:c�(v)=k d(v). But since each degree d(v) is bounded by d, then as shown in [7] this



can be determined in polynomial time, as required. Thus the whole partition conditioncan be tested in polynomial time. �

4. ACHROMATIC NUMBER

We now turn our attention to the achromatic number of graphs in various classes. Wefirst quote a theorem from [7] which gives a useful equivalent of the degree partitionextension condition.

Theorem 4.1. Let G be a graph with maximum degree d, m=(r2

)edges, and ni

vertices of degree i and let S={i :ni ≥ (d−1)(r−1)}. Then for sufficiently large r, Gsatisfies the degree partition extension condition if and only if there exists a partialharmonious coloring c0 of the rare vertices and an arbitrary (not necessarily proper)extension c of c0 so that for each color class c−1(i), the degree sum ds(c−1(i),G) isequivalent to r−1modgcd(S).

Using this, we can obtain the following lower bound on the achromatic number ofsufficiently large graphs:

Theorem 4.2. Let d be a fixed positive integer, and let � be a fragmentable class oftriangle free graphs of maximum degree at most d. Let G∈�, with m edges, and for eachi, let ni be the number of vertices of degree i in G. Let T ={i :ni ≥ (2d−1)(q(m)−1)},and set g=gcd(T). Then the achromatic number �(G) is at least q(m)−�(g−1) /2�provided m is large enough.

Proof. Set k=�(g−1) /2� and let r=q(m)−k. We will show that G has a completecoloring with r colors. Let D be the total degree excess in G, i.e. the amount by whichthe total degree exceeds r(r−1). Thus

D = 2

[m−

(r

2

)]

= 2

[(q(m)

2

)+r(m)−

(r

2

)]

≥ 2

[(q(m)

2

)−(

q(m)−k

2

)]= 2kq(m)−(k+1)k

≥ 2�(g−1) /2�q(m)−(k+1)k.

Hence D≥ (g−1)r if

2�(g−1) /2�q(m)−(k+1)k≥ (g−1)(q(m)−k)

i.e. if

(2�(g−1) /2�−(g−1))q(m)+(g−1)k≥ (k+1)k.



For g even, this is clearly true if m is large enough. If g is odd, then g=2k+1, soD≥ (g−1)r provided 2k2 ≥k2 +k, which is true for all k≥0. Hence D≥ (g−1)r. Notethat D is even.

Now, as in the proof of Theorem 3.3, we can find a set of O(n) edges which are ata distance at least 3 apart from each other and from any scarce degree vertex. Sincewe have O(n) such edges and there are only d different vertex degrees, we can choosea subset I of 3D /2 edges such that all of the edges in I have endpoints with degreesd0, d1 say (where d0, d1 may be equal). Partition I into D /2 sets each containing threeedges.

Now for each set of three edges, say v0v1, v2v3, v4v5, delete the edge v2v3, andidentify the vertices in pairs {v0,v5}, {v1,v2}, {v3,v4}. This creates an exact (simple)graph G′ with

(r2

)edges. Since D /2≥ 1

2 (g−1)r, it follows that neither of the degreesd0 +d1,d0 +d1 −1 will be rare in G′ provided r is large enough.

For each j, let n′j be the number of vertices of degree j in G′, and let d′ ≤2d be the

maximum degree of G′. Let S={j :n′j ≥ (d′−1)(r−1)}, and set g′ =gcd(S). Since the

only degrees which are less common in G′ than in G are d0,d1, and there are O(n)vertices with each of these degrees, then it follows that S⊇T , and hence g′ ≤g (in factg′|g).

Now color each rare degree vertex of G′ with a distinct color; this gives a partialharmonious coloring c0 of G′. We will extend c0 to a (not necessarily proper) coloring cof a subset of V(G′) such that for each color i, the degree sum ds(c−1(i),G) is equivalentto r−1modg′. To do this, choose, for each i, a non-negative integer ti ≤g′−1≤g−1such that ds(c−1

0 (i),G)+ ti ≡r−1modg′.Since there are at least (g−1)r vertices of degree d0 +d1 −1 and O(n) vertices of

degrees d0 and d1, we can, for each i, pick g′− ti (or 0 if ti =0) vertices of degreed0 +d1 −1 and ti vertices of degree d0 and of degree d1, and color these i. Then wehave

ds(c−1(i),G) = (g′− ti)(d0 +d1 −1)+ tid0 + tid1 +ds(c−10 (i),G)

≡ ti +ds(c−10 (i),G)modg′

≡ r−1modg′.

Now by Theorem 4.1, G′ satisfies the degree partition extension condition. Henceby Theorem 3.1, G′ has an exact coloring with r colors, which immediately givesa complete coloring of G with r colors. Hence �(G)≥r=q(m)−�(g−1) /2�, asrequired. �

Remark. For each d, it is possible to construct fragmentable classes of triangle-free graphs which are regular of degree d, with m edges and m arbitrarily large, suchthat �(G)≤q(m)−�(d−1) /2�=q(m)−�(g−1) /2�. Hence the bound of Theorem 4.2cannot be improved in general.

The construction varies slightly according to the value of d modulo 8. Let d=8a+k,where 0≤k≤7. It will be useful to list five quantities �k, �k, �k, �k, �k for each



k=0, . . . ,7; these are given in the table below.

k 0 1 2 3 4 5 6 7

�k 1 2 2 3 3 3 4 4�k 2a 3a 0 a 2a+1 3a+2 0 a+1�k b+1 2b+2 b+1 2b+2 b+2 2 b+2 2�k 2 3 2 3 2 2 2 2�k 0 0 1 1 2 2 3 3

For any positive integer b, set q=db+4a+�k. Also set r=�k. Let m=(q2)+r, andlet n=2m /d. Then straightforward calculations show that in each case n is an integer,in fact n=db2 +db+2a+�k. Note that for each d, there are infinitely many values ofb for which �k, and therefore n, is even. Then we can construct a d-regular bipartitegraph Bn,d on n vertices as follows: Let V(Bn,d)=X∪Y , where |X|=|Y|=n /2, and letX ={x0 . . .xn/2−1}, Y ={y0 . . .yn/2−1}. Join xi to yj if (i− j) mod (n /2)∈{±1, . . . ,±�d /2�}; also, if d is odd, join each pair xi,yi. As n varies, this gives an infinite family oftriangle-free d-regular graphs which is fragmentable. Note that |E(Bn,d)|=m.

Note that since r<q, we have q(m)=q, so that �(Bn,d)≤q=db+4a+�k. Nowconsider a complete coloring of Bn,d. Then either (i) some color has degree sum atmost db, or (ii) every degree sum is at least db+d.

In case (i), we must have �(Bn,d)≤db+1. In case (ii), note that it is easily checkedthat if b>2a, then 2m=nd<(db+d)(db+�k). Hence since every degree sum is atleast db+d, the number of colors is at most 2m / (db+d)<db+�k, so that �(Bn,d)≤db+�k −1.

Hence in either case we have

�(Bn,d)≤db+�k −1=q(m)−(4a+�k −�k +1)=q(m)−(4a+�k),

and 4a+�k =�(d−1) /2�. So finally we have

�(Bn,d)≤q(m)−⌈

d−1

2

⌉

for n large enough, as required.Since every sufficiently large graph of some fixed genus has many vertices of degree

at most 6, we get the following:

Theorem 4.3. Let d, � be fixed non-negative integers, and let �d,� be the class ofgraphs of maximum degree d and genus �. Then if G∈�d,� is triangle-free and has medges, and m is large enough, �(G)≥q(m)−3.

Proof. Apply Theorem 4.2, noting that g≤6. �



From Theorem 4.2 and Lemma 3.5, we obtain a polynomial time algorithm:

Theorem 4.4. Let d be a fixed positive integer, and let � be a fragmentable classof triangle-free graphs of maximum degree at most d. Then for G∈�, the achromaticnumber �(G) can be determined in polynomial time.

Proof. From Theorem 4.2 we know that �(G)≥q(m)−�(d−1) /2�. Then for each rwith q(m)−�(d−1) /2�≤r≤q(m), we know that if m is large enough G has a completecoloring with r colors if and only if G satisfies the partition condition with r parts; thegreatest r for which this holds is �(G).

But for each r, we can test in polynomial time whether or not G satisfies the partitioncondition with r parts, by Lemma 3.5.

If m is not large enough for Theorem 4.2 to apply, then we can (at least in principle)use exhaustive search to determine �(G). Thus the whole algorithm runs in polynomialtime. �

Theorem 4.5. Let D≥2 be a fixed integer, and let LD(N) be the N×·· ·×N︸︷︷︸D

lattice

graph with vertex set {1, . . . ,N}D and two vertices (x1, . . . ,xD), (y1, . . . ,yD) adjacent ifand only if

∑i |xi −yi|=1. Then if N is large enough �(LD(N))=q(m), where m is the

number of edges of LD(N).

Proof. Note that the lattices are fragmentable [8] and triangle free, so we needonly check that they satisfy the partition condition with q(m) parts.

For D≥3 we use Theorem 4.2. Let n be the number of vertices of LD(N), then n=ND.It is easily seen that LD(N) has (N−2)D vertices of degree 2D, and 2D(N−2)D−1

vertices of degree 2D−1. It is then immediate from Theorem 4.2 that �(LD(N))=q(m)provided N is large enough.

For the case D=2, L2(N) has m=2N(N−1) edges, so that q(m)=2N−1 and r(m)=N−1. Now L2(N) has 4 vertices of degree 2, 4N−8 of degree 3, and N2 −4N+4 ofdegree 4. We apply Theorem 3.3 with r=q(m). Take P=SG and Edel =∅. Now if N isodd, r−1=q(m)−1=2N−2≡0mod4. The only scarce degree vertices are those ofdegree 2 (for large enough N), and since these are at the corners of the lattice, we candefine c� by taking these to be all of the same color. Then we can obtain the partition� by grouping together the degree 3 vertices into groups of 4, and then successivelyadding these groups, followed by the degree 4 vertices, to the part with the least currentdegree sum. Since the total degree excess is 2N−2, some parts will end up with degreeexcess 0, and the remainder (at least 2) with degree excess 4.

If N is even, r−1=q(m)−1=2N−2≡2mod4. This time, color the degree 2vertices with distinct colors. Then we can form q(m)−4=2N−5 pairs of degree threevertices, and place one pair into each of the remaining parts so each part now hasdegree sum 2 or 6. Then place the degree 4 vertices (and finally the remaining twodegree 3 vertices) successively into the part with least current degree sum. As before,some parts will end up with degree excess 0, some with degree excess 4, and two withdegree excess 3.



In either case, it is easy to see that the partition satisfies conditions (P1)–(P4) ofTheorem 3.3, and so L2(N) has a complete coloring with r colors. �

Remarks. By similar arguments, it can be shown that if NM is large enough,with N,M ≥2, then any N×M lattice L2(N×M) with m edges has �(L2(N×M))=q(m) if q(m)≡0,1,3mod4. However if q(m)≡2mod4, there exist N,M such that�(L2(N×M))=q(m)−1. For example, for any r≥2, we may take Nr =2r2 −3r+2

and Mr =2r2 −r+1. Then m=2NrMr −Nr −Mr =(4r2−4r+22

). Thus L2(Nr ×Mr) is an

exact graph and so a complete coloring with q(m)=4r2 −4r+2 colors must have eachdegree sum equal to 4r2 −4r+1, i.e. congruent to 1 modulo 4. However, it is not hardto see that there are insufficient vertices of degree 3 to make this possible.

A. Cubic Planar Triangle-Free Graphs

As a further example, we consider the case of cubic planar triangle-free graphs. In thiscase, provided m is large enough, there can be no vertices of scarce degree, and sincethe graph is regular, the bin packing problem is very simple, and gives the following:

Theorem 4.6. Let G be a cubic planar triangle-free graph with m edges. Then if mis sufficiently large, the achromatic number �(G) of G is given by

�(G)=

⎧⎪⎪⎨⎪⎪⎩

�√

2m� if q(m)≡0mod3

q(m) if q(m)≡1mod3

q(m)−1 if q(m)≡2mod3.

5. CONCLUDING REMARKS

For any bounded degree class of graphs, the partition condition on r parts is a necessarycondition, testable in polynomial time, for there to exist a complete coloring with rcolors. We conjecture that this condition is also sufficient (for sufficiently large graphs)in wider classes of graphs than those considered here, perhaps even for all boundeddegree classes. Also, the methods used here only allow us to prove the conditionsufficient for extremely large graphs, but it seems likely that it remains sufficient formuch smaller graphs.

APPENDIX A

We prove here a technical lemma which allows us to pick elements of a set of sequenceswhich are, in a suitable sense, much bigger than the earlier terms, or to obtain boundson the size of the elements. Denote the non-negative real numbers by R+.

Lemma A.1. Let fi :Rk+ →R+, i=1, . . . ,k, be increasing functions. Then there is a

function F : (N∪{−1})k →R+ such that the following holds Let {a(i)n }n≥0, i=1, . . . ,k,



be k sequences of non-negative real numbers. For any N1, . . . ,Nk ≥0, either (a) thereexist non-negative integers M1 ≤N1, . . . ,Mk ≤Nk such that for each i=1, . . . ,k,

a(i)Mi

≥ fi

( ∑j<M1

a(1)j , . . . ,

∑j<Mk

a(k)j

)

and ∑j<Mi

a(i)j ≤F(M1 −1, . . . ,Mk −1)

or (b) there is an i∈{1, . . . ,k} such that∑j≤Ni

a(i)j ≤F(N1, . . . ,Nk).

Proof. First we define the function F. If (M1, . . . ,Mk)∈ (N∪{−1})k with Mt ≥0, itis useful to write F(Mt) to mean F(M1, . . . ,Mt −1, . . . ,Mk). We define F as follows:

F(−1, . . . ,−1) = 0,

F(M1, . . . ,Mk) = maxt:Mt≥0

{F(Mt)+ ft(F(Mt), . . . ,F(Mt))} otherwise.

We will construct a digraph on vertex set V = (N∪{−1})k as follows: If

a(i)Mi

<fi

( ∑j<M1

a(1)j , . . . ,

∑j<Mk

a(k)j

),

then there is an edge from (M1, . . . ,Mi −1, . . . ,Mk) to (M1, . . . ,Mi, . . . ,Mk); there are noother edges.

Let B be the set of vertices (M1, . . . ,Mk) such that there is a path from vertex(−1, . . . ,−1) to (M1, . . . ,Mk). We claim that if (M1, . . . ,Mk)∈B, then for each i,∑

j≤Mi

a(i)j ≤F(M1, . . . ,Mk).

We prove this by induction on S=∑ki=1 Mi. If S=−M, then M1 =·· ·=Mk =−1. Then

we just need to show that for each i,

0= ∑j≤−1

a(i)j ≤F(−1, . . . ,−1)

which follows from the definition of F.If S>−M, then if (M1, . . . ,Mk)∈B, there must be an edge from some vertex

(M1, . . .Mt −1, . . . ,Mk)∈B to (M1, . . .Mt, . . . ,Mk). Set (M′1, . . . ,M′

k)= (M1, . . .Mt −1, . . . ,Mk). Since there is an edge from (M1, . . .Mt −1, . . . ,Mk) to (M1, . . .Mt, . . . ,Mk),we have

a(t)Mt

<ft

( ∑j<M1

a(1)j , . . . ,

∑j<Mk

a(k)j

)



and since (M′1, . . . ,M′

k)∈B, we have, for each i,∑j≤M′

i

a(i)j ≤F(M′

1, . . . ,M′k)=F(Mt).

If i �= t, then we already have∑j≤Mi

a(i)j ≤F(Mt)≤F(M1, . . . ,Mk).

If i= t, then we have ∑j≤Mt−1

a(t)j ≤F(Mt).

But we also have

a(t)Mt

<ft

( ∑j<M1

a(1)j , . . . ,

∑j<Mk

a(k)j

)

and for each i, ∑j<Mi

a(i)j ≤F(Mt).

Hence

∑j≤Mt

a(t)j ≤ F(Mt)+ ft

( ∑j<M1

a(1)j , . . . ,

∑j<Mk

a(k)j

)

≤ F(Mt)+ ft(F(Mt), . . . ,F(Mt))

≤ F(M1, . . . ,Mk)

as required.Now suppose there is a vertex (M1, . . . ,Mk)∈B with Mi =Ni for some i, then (b)

holds since we have ∑j≤Ni

a(i)j ≤F(N1, . . . ,Nk).

On the other hand, suppose there is no such vertex. Choose (M1 −1, . . . ,Mk −1)∈B suchthat

∑i Mi is a maximum. Then 0≤Mi ≤Ni for each i. Now for each i, we know that

(M1 −1, . . . ,Mi, . . . ,Mk −1) �∈B, so there cannot be an edge from (M1 −1, . . . ,Mk −1) to(M1 −1, . . . ,Mi, . . . ,Mk −1), i.e.

a(i)Mi

≥ fi

( ∑j<M1

a(1)j , . . . ,

∑j<Mk

a(k)j

).

Also, for each i, ∑j≤Mi−1

a(i)j ≤F(M1 −1, . . . ,Mk −1)

so that (a) holds, as required. �



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Achromatic number of fragmentable graphs