Decompositions of complete graphs into long cycles

Abstract

The problem of decomposing complete graphs into cycles of arbitrary specified lengths has attracted much attention, but remains largely unsolved. In this paper, the problem is settled in the case where the specified cycle lengths are each more than about half the order of the complete graph. The proof is based on a result that modifies certain existing cycle decompositions to produce new ones in which the lengths of two of the cycles are altered.

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].…”

“…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.…”

“…Our aim in this section is to prove the following lemma, which was proved for the case = 1 in [13]. To accomplish this we will prove Lemma 3.5, from which Lemma 3.1 follows immediately.…”

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 shown in [5] that there exists an (X) * -decomposition of K n for each n-admissible list X such that each entry of X is at least We shall now begin with D † and repeatedly apply Lemma 2.1, on each occasion choosing y to be the largest legal integer and x to be the smallest legal integer, where a legal integer is an integer which is the length of a cycle in the current decomposition and is not e unless there are at least two cycles of length e in the current decomposition. Noting the following points it is clear that we can follow this procedure until we obtain the required (M, e) * -decomposition of K n .…”