The Hamilton–Waterloo Problem with even cycle lengths

Abstract

The Hamilton-Waterloo Problem HWP(v; m, n; α, β) asks for a 2factorization of the complete graph K v or K v − I, the complete graph with the edges of a 1-factor removed, into α C m -factors and β C nfactors, where 3 ≤ m < n. In the case that m and n are both even, the problem has been solved except possibly when 1 ∈ {α, β} or when α and β are both odd, in which case necessarily v ≡ 2 (mod 4). In this paper, we develop a new construction that creates factorizations with larger cycles from existing factorizations under certain conditions. This construction enables us to show that there is a solution to HWP(v; 2m, 2n; α, β) for odd α and β whenever the obvious necessary conditions hold, except possibly if β = 1; β = 3 and gcd(m, n) = 1; α = 1; or v = 2mn/ gcd(m, n). This result almost completely settles the existence problem for even cycles, other than the possible exceptions noted above.

The Hamilton-Waterloo problem on wreath product graph $C_m \wr K_{16}$

“…For the following lemmas, we need two special 1-factorizations of K 16 with the vertex-set Z 16 whose 15 1-factors are listed as below. (3,6), (4, 5), (7,10), (8,9), (11,14), (12,13), (15, 2)}, (5,8), (6,7), (9,12), (10,11), (13, 0), (14,15), (1, 4)}, I3 = {(0, 2), (6, 1), (13, 3), (7,9), (5,11), (15,12), (8,10), (14, 4)}, (10,5), (1,7), (11,13), (9,15), (3, 0), (12,14), (2, 8)}, I5 = {(0, 4), (10, 1), (11,3), (9, 2), (12,8), (14,5), (15,7), (13,6)}, I6 = {(6, 10), (0, 7), (1,9),...…”

“…Theorem 1.1. There exists a C k -factorization of K u [g] if and only if g(u − 1) ≡ 0 (mod 2), gu ≡ 0 (mod k), k is even when u = 2, and (k, u, g) ∈ {(3, 3, 2), (3,6,2), (3,3,6), (6,2,6)}.…”

“…Most of the known results on the Hamilton-Waterloo problem for small values of m and n. A complete solution for the existence of an HW(v; 3, n; α, β) in the cases n ∈ {4, 5, 7} is given in [1,4,8,12,15,19]. The cases (m, n) ∈ { (3,15), (5,15), (4,6), (4,8), (4,16), (8,16)} are solved in [1]. The existence of an HW(v; 4, n; α, β) for odd n ≥ 3 has been solved except possibly when v = 8n and α = 2, see [11,15,19].…”

“…The authors [3] have constructed many infinite classes of HW(v; 3, 3x; α, β)s. In [18], the present authors consider the Hamilton-Waterloo problem with C 8 -factors and C m -factors. For more results on the Hamilton-Waterloo problem, the reader can see [5,6,20].…”

On the generalized Oberwolfach problem

“…Many more existence results are known when t = 2, that is, for HWP(K * v ; α 1 F 1 , α 2 F 2 ) although they mostly concern the uniform case, which is nonetheless still open. Some recent methods that allowed progress in the uniform case [27,28,29,31,33] will be discussed in Section 2.1.…”