Decompositions of complete graphs into triangles and Hamilton cycles

Abstract

For all odd integers n ! 1, let G n denote the complete graph of order n, and for all even integers n ! 2 let G n denote the complete graph of order n with the edges of a 1-factor removed. It is shown that for all non-negative integers h and t and all positive integers n, G n can be decomposed into h Hamilton cycles and t triangles if and only if nh þ 3t is the number of edges in G n . #

Hamilton cycle rich two‐factorizations of complete graphs

“…[9], is an easy consequence of Lemma 2.1. The following lemma, proved independently by Rodger [20], is the key ingredient needed for the proofs of our main results.…”

Cycle decompositions of complete multigraphs

“…Of course, there have been numerous articles devoted to (M) * -decompositions of K n when = 1. Most of these, for example see [1,5,6,12,16,18,20,21], are cited in the survey [10], but some additional results have been obtained since [10] appeared. In particular, it is shown in [14] that for all sufficiently large odd n, there is an (M) * -decomposition of K n for each (1, n)-admissible list M. Results similar to Theorems 1.2 and 1.3 are proven for = 1 in [13,14], respectively.…”

An asymptotic solution to the cycle decomposition problem for complete graphs

“…In recent work, the authors have settled the problem for about 10% of all n-admissible lists (see [10]), and for the case where all the cycle lengths are at least about n 2 (see [11]). Numerous other results on the problem can be found in [1,5,13,[17][18][19]. For further results on cycle decompositions, and on graph decompositions generally, see the surveys [15] and [8].…”

“…Our constructions for this purpose follow the basic outline of the idea which was used in [13] to find decompositions of complete graphs into all admissible combinations of 3-cycles and Hamilton cycles. It would be nice to apply similar techniques to the related cycle decomposition problem which arises for even values of n, but our constructions rely on the fact that small powers of 2 are relatively prime to n when n is odd, and so there is no apparent way to do this.…”