Packing cycles in complete graphs

Abstract

We introduce a new technique for packing pairwise edge-disjoint cycles of specified lengths in complete graphs and use it to prove several results. Firstly, we prove the existence of dense packings of the complete graph with pairwise edge-disjoint cycles of arbitrary specified lengths. We then use this result to prove the existence of decompositions of the complete graph of odd order into pairwise edge-disjoint cycles for a large family of lists of specified cycle lengths. Finally, we construct new maximum packings of the complete graph with pairwise edge-disjoint cycles of uniform length.

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.…”

“…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.…”

Maximum packings of the complete graph with uniform length cycles

“…We also require two results from [4]. Lemma 3.1 is an amalgam of Lemmas 3.5 and 3.6 of [4], and Lemma 3.2 is adapted from Lemma 4.14 of [4].…”

“…From these decompositions it is easy to obtain the packings that we require for Theorem 1.1, except in the case where n is even and n 2 −km ∈{ n 2 +1, n 2 +2}. In Section 3, we show that in this remaining case we can still obtain the required packings from the decompositions given by Lemma 2.5 by using two lemmas from [4].…”

Constructing the Spectrum of Packings and Coverings for the Complete Graph with Stars with up to Five Edges

“…In 1999, Adams, Bryant, and El-Zanati [2] solved the packing and 1 covering problems for 3-cubes. In 2008, Bryant and Horsley [4] found sufficient conditions for the existence of a packing of the complete graph with cycles of specific lengths. Bryant [3] also proved Tarsi's conjecture [15] on necessary and sufficient conditions for the existence of a packing of the complete multigraph with paths of specific lengths in 2010.…”