An asymptotic solution to the cycle decomposition problem for complete graphs

Abstract

Let m 1 , m 2 , . . . ,m t be a list of integers. It is shown that there exists an integer N such that for all n N, the complete graph of order n can be decomposed into edge-disjoint cycles of lengths m 1 , m 2 , . . . ,m t if and only if n is odd, 3 m i n for i = 1, 2, . . . ,t, and m 1 +m 2 + · · · +m t = n 2 . In 1981, Alspach conjectured that this result holds for all n, and that a corresponding result also holds for decompositions of complete graphs of even order into cycles and a perfect matching.

Cycle decompositions of complete multigraphs

“…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. For further results on cycle decompositions or on graph decompositions generally, see the surveys [11,17].…”

“…A corresponding (though somewhat stronger) result has been proven for the case = 1 in [14]. We begin with some observations and definitions.…”

“…This article uses similar edge swapping techniques to those introduced in [15] and used in [12][13][14]. In the following lemma, we extend the technique to deal with packings of K n for >1.…”

DecomposingKu+w−Kuinto cycles of prescribed lengths

“…This 'switching' method was first applied to packings of the complete graph [3][4][5], and has since been generalised to other graphs [6,12,14]. See [13] for a proof of Lemma 3 and a survey of switching techniques for graph decompositions.…”

Maximum packings of the complete graph with uniform length cycles

“…It is clear that e ≤ m+2, so by calculating the values of e when m ∈{n−1, n} (noting (n, m) / ∈ {(5, 4), (6, 5)}) it can be seen that e ≤ n. Thus, since Balister [2] has shown that there exists an (X) * -decomposition of K n for every n-admissible list X when n ≤ 14, we may assume that n ≥ 15. Let k = It is shown in [6] that there exists an (X) * -decomposition of K n for each n-admissible list X such that the largest entry of X is at most n+1 2 and the second largest entry of X is at least half the largest. Thus, since e ≤ m+2 and k is obviously at least 2, it suffices to show that e ≤ 2 ), we obtain the required (M, e) * -decomposition of K n .…”