On the Hamilton–Waterloo problem with odd cycle lengths

Abstract

Let Kv∗ denote the complete graph Kv if v is odd and Kv−I, the complete graph with the edges of a 1‐factor removed, if v is even. Given nonnegative integers v,M,N,α,β, the Hamilton–Waterloo problem asks for a 2‐factorization of Kv∗ into α CM‐factors and β CN‐factors, with a Cℓ‐factor of Kv∗ being a spanning 2‐regular subgraph whose components are ℓ‐cycles. Clearly, M,N≥3, M∣v, N∣v and α+β=⌊v−12⌋ are necessary conditions. In this paper, we extend a previous result by the same authors and show that for any odd v≠MNgcd(M,N) the above necessary conditions are sufficient, except possibly when α=1, or when β∈{1,3}. Note that in the case where v is odd, M and N must be odd. If M and N are odd but v is even, we also show sufficiency but with further possible exceptions. In addition, we provide results on 2‐factorizations of the complete equipartite graph and the lexicographic product of a cycle with the empty graph.

The Hamilton–Waterloo Problem with even cycle lengths

“…In the remainder of this section, we will solve HWP(C 2m/g [n]; 2m, 2n; α, β), except possibly when β ∈ {1, 3}, or α = 1 and g > 1. We first recall the following result from [11]. In the case g = 1, however, we can improve this result, removing many of the exceptions and allowing the possibility that m is even in some cases.…”

“…[11]). Let m and n be odd integers with n > m ≥ 3, let g = m be a common divisor of m and n, and let α and β be nonnegative integers.…”

On the Hamilton–Waterloo Problem with cycle lengths of distinct parities

“…Theorem 2.4 (Theorem 2.9, [12]). Let n be a positive integer, and let g and g ′ be positive divisors of n. Also, let T be a subset of Z n and ℓ ≥ 3.…”

“…Note that the above Lemma has been independently proven in [18] with different techniques. An alternative proof in the case where M is odd can be found in [12].…”

On the generalized Oberwolfach problem

“…Some of the most important results up to 2006 can be found in [15,Section VI.12]. More recent results can be found in [4,7,8,10,12,13,16,23,25]. For more details and some history on the problem, we refer the reader to [12,13].…”