The Hamilton—Waterloo problem: The case of triangle‐factors and one Hamilton cycle

Abstract

The Hamilton-Waterloo problem is to determine the existence of a 2-factorization of K 2n+1 in which r of the 2-factors are isomorphic to a given 2-factor R and s of the 2-factors are isomorphic to a given 2-factor S, with r +s = n. In this paper we consider the case when R is a triangle-factor, S is a Hamilton cycle and s = 1. We solve the problem completely except for 14 possible exceptions. This solves a major open case from the 2004 paper of Horak, Nedela, and Rosa.

A note on the Hamilton–Waterloo problem with C8-factors and CmWang1, Cao2 2018Discrete Mathematics90100View full textAdd to dashboardCite

“…We can get the conclusion by using Construction 3.1 with an HW(8; 8, m; 3, 0) and an HW(K mt [8] It is similar to the first case, P mt−2…”

“…C m [8] can be partitioned into r 1 C 8 -factors and 8 − r 1 C m -factors for m ≥ 3 and r 1 ∈ {0, 2, 4, 8} by Lemma 3.2. Since K 8 can be decomposed into three C 8 -factors and a 1-factor by Theorem 1.1, the graph mK 8 can be decomposed into three C 8 -factors and a 1-factor.…”

“…For the case (m, n) ∈ {(3, 15), (5,15), (4,6), (4,8), (4,16), (8,16)}, see [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 [13,21,24].…”

“…Many infinite classes of HW(v; 3, 3x; α, β)s are constructed in [3]. For more results on the Hamilton-Waterloo problem, the reader can see [5,8,11,15,16,25]. In this paper, we focus on the existence of an HW(8mt; 8, m; α, β).…”

“…∪ mtK 8 can be decomposed into r C 8 -factors, 11 − r C m -factors and a 1-factor on the vertex set Z mt × Z 8 for m ≥ 3 and 3 ≤ r ≤ 11. mt 2 K 2 [8] can be partitioned into 4 C 8 -factors by Lemma 1.1. We together get α = 8x+r+4 C 8 -factors, β = 8·( mt−4 2 −x)+11−r = 4mt−1−(8x+r+4) C m -factors and a 1-factor for 0 ≤ x ≤ mt−4 2 and 3 ≤ r ≤ 11.…”