Resolvable Group Divisible Designs with Block Size 3

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

“…A -factorization of is a partition of ( ) into -factors. Many authors contributed to proving the following known result, see [4,6,25,34,35,41,42].…”

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

“…A factor of H is a spanning subgraph of H. Suppose G is a subgraph of H, a G-factor of H is a set of edge-disjoint subgraphs of H, each isomorphic to G. A G-factorization of H is a set of edge-disjoint G-factors of H. A C k -factorization of H is a partition of E(H) into C k -factors. Many papers introduced C k -factorization of K u [g], see [2,4,10,18,19,20,22,23]. Theorem 1.1.…”

“…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 Hamilton-Waterloo Problem for C3-Factors and Cn-Factors

“…We denote the cycle of length k by C k , the complete graph on v vertices by K v , and the complete u-partite graph with u parts of size g by K u [g]. A factor of a graph H is a spanning subgraph of H. Suppose G is a subgraph of a graph H, a G-factor of H is a set of edge-disjoint subgraphs of H, each isomorphic to G. And a G-factorization of H is a set of edge-disjoint G-factors of H. Many authors [2,4,15,16,18,19,25,26] have contributed to prove the following result. Theorem 1.1.…”

“…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)}.…”