Isomorphic Decomposition of Complete Graphs into Linear Forests

Sergio Ruiz wEsrmN MICHIGAN uNivERsirY AND

UNIV€RSIDAD CATOLICA DE VALPARAiSO

ABSTRACT

The well-known theorem of Kirkman states that every complete graph Kzn of order 2n is I-factorable or, equivalently, is nKz-decomposable. This result is generalized to any linear forest of size n without isolated vertices.

By a decomposition of a graph G with at least one edge is meant a family of subgraphs G I , G2, ..., Gk of G such that their edge sets form a partition of the edge set of G. Any member of the family is called a parr (of the decomposition).

A graph G is said to be H-decomposable (or has an H-decomposition) if G has a decomposition in which all its parts are isomorphic to the graph H . We also say that G has an isomorphic decomposition into the graph H . An obvious necessary condition for a graph G to be H-decom- posable is that the size of G is a multiple of the size of H.

In 1947, Kirkman [ I ] discovered a combinatorial theorem which, when expressed in graph theory terminology, stipulates that the complete graph K2, is 1-factorable, or equivalently, K2,, is nK2-decomposable for every n 3 1. It is the object of this paper to generalice Kirkman’s Theorem to linear forests (graph each component of which is a path).

Theorem. If F is a linear forest of size n without isolated vertices, then K2, is F-decomposable.

Proof. Since the result is obvious for n = I , we assume that n 3 2. Let uo, u I , ..., u2 ,,-, be the vertices of K2,#. Arrange the vertices u l , u 2 ,

Journal of Graph Theory, Vol 9 (1985) 189-191 C 1985 by John Wiley & Sons, Inc CCC 0364-9024/85/010189-03804 00

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..., u ~ ~ - ~ cyclically, around vo as center, in counterclockwise order about a regular (2n - 1)-gon. Next, we define the length I ( v j v j ) of an edge vjvj as

m i n { l i - j J , ( 2 n - I - i + j l } i f l < i , j s 2 n - I L(UjVj ) = if either i or j is equal to 0.

Note that for each i E (0, 1, ..., n - l}, our representation of K2, contains 2n - 1 edges of length i. It follows that by selecting n edges of distinct lengths, an isomorphic part H of Kz, is induced by these edges. The other parts of the decomposition are obtained from H by the consecutive powers of the permutation

or, equivalently, in our geometrical model, by counterclockwise rotations around uo in an angle of 27r/(2n - 1) radians. With this background at hand, it suffices to show that the subgraph H can be chosen so that it is isomorphic to the given linear forest F . We now describe two paths P and Q of length n in K2n. If n is even, then

and

while if n is odd, then

and

Observe that, in either case, the ith edge of P and the ith edge of Q have length i - 1 for i = 1, 2, ..., n.

Assume that the linear forest is given by

where then n, + n2 + ... + nk = n . Finally a subgraph H isomorphic to F is induced by the following edges of K:,,: the first n , edges of P , edges n, + 1 through n l + n2 of Q , edges n, + n2 + 1 through n, +

DECOMPOSING K2c INTO LINEAR FORESTS 191

P "1

FIGURE 1. A step in the construction of an F-decomposition of KI2 for F = P2 u f 3 u P4.

nz + n3 of P, etc. until the last nA edges of Q are taken if k is even or the last n, edges of P if k is odd. The edges chosen have distinct lengths by the definition of P and Q ; thus the proof is complete.

The preceding theorem and its proof are illustrated in Figure 1 for 2n = 12 and F = P2 U P3 U P , . The labeling of the vertices of K 1 2 is shown along with the edges that induce the initial copy of F.

Corollary 1 (Kirkman). The complete graph K z , is 1-factorable.

Corollary 2. The complete graph K2,,-, is F-decomposable for any linear forest F of size n ( 2 1 ) and without isolates.

By removing the central vertex uo in the proof of the Theorem along with the first edge in the paths P and Q , the same construction applies, except that now we do not have edges of length 0.

Reference

[ 11 T. P. Kirkman, On a problem in combinations. Cambridge andDublin Murh. J . 2 (1847) 191-204.