Further results on almost resolvable cycle systems and the Hamilton–Waterloo problem

Abstract

In this paper, we almost completely solve the existence of almost resolvable cycle systems with odd cycle length. We also use almost resolvable cycle systems as well as other combinatorial structures to give some new solutions to the Hamilton-Waterloo problem. K E Y W O R D Salmost resolvable cycle system, Hamilton-Waterloo problem, 2-factorization

On the Hamilton–Waterloo problem with odd cycle lengths

“…For the case , we recall a result proven in [, Theorem and Lemma 3.15]. We point out that the exceptions , (3, 13, 8), (3, 15, 8) in [, Theorem 1.6] have been dealt with in . We state this result as follows.…”

“…If , or , or , (3, 13, 8), (3, 15, 8)}, we apply Theorem . Finally, the cases , (3, 13, 8), (3, 15, 8) have been proven in . If or 5, then the result comes from Lemmas and , respectively.…”

Almost resolvable k‐cycle systems with

“…In this section, we present some preliminary notation and definitions, and provide lemma for the construction of a k ‐ARCS for . We point out that similar methods have been used for many years (see ).…”

“…) presented the following open problem “The outstanding problem remains the construction of almost resolvable 2 k ‐cycle systems of order , since this will determine the spectrum for almost resolvable 2 k ‐cycle systems with the one possible exception of orders .” Since then, many authors have contributed to proving the following known conclusions. Theorem () Let . There exists a k ‐ARCS( n ) for any odd and any even , except for and except possibly for .…”

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

“…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).…”