The Hamilton-Waterloo Problem for C3-Factors and Cn-Factors

Abstract

The Hamilton-Waterloo problem asks for a 2-factorization of K v (for v odd) or K v minus a 1-factor (for v even) into C m -factors and C n -factors. We completely solve the Hamilton-Waterloo problem in the case of C 3 -factors and C n -factors for n = 4, 5, 7.

On the Hamilton–Waterloo problem with odd cycle lengths

“…Much of the literature on the Hamilton–Waterloo problem seeks to solve the problem for specific cycle lengths M and N . The case was mostly solved in , with the remaining exceptions solved in . The cases are solved in , except that left open for .…”

“…The cases are solved in , except that left open for . This case was recently solved in . For , a near‐complete solution was given in , with the remaining exceptions solved in .…”

A note on the Hamilton–Waterloo problem with C8-factors and CmWang1, Cao2 2018Discrete MathematicsSelf Cite90100View full textAdd to dashboardCite

“…The Hamilton-Waterloo problem is the problem of determining whether K v (for v odd) or K v minus a 1-factor (for v even) has a 2-factorization in which there are exactly α C m -factors and β C n -factors. The authors [24] generalize this problem to an r-regular graph H, and use HW(H; m, n; α, β) to denote a 2-factorization of H (for r even) or H minus a 1-factor (for r odd) in which there are exactly α C m -factors and β C n -factors. Denote by HWP(H; m, n) the set of (α, β) for which an HW(H; m, n; α, β) exists.…”

“…Many authors have considered 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 [1,7,14,21,24]. For the case (m, n) ∈ {(3, 15), (5,15), (4,6), (4,8), (4,16), (8,16)}, see [1].…”

“…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]. The authors [9] give a complete solution for the existence of an HW(v; 4, n; α, β) for even n ≥ 4.…”