Fair holey hamiltonian decompositions of complete multipartite graphs and long cycle frames

Abstract

a b s t r a c tA k-factor of a graph G = (V (G), E(G)) is a k-regular spanning subgraph of G. A k-factorization is a partition of E(G) into k-factors. Let K (n, p) be the complete multipartite graph with p parts, each of size n. If V 1 , . . . , V p are the p parts of V (K (n, p)), then a holey k-factor of deficiencyHence a holey k-factorization is a set of holey k-factors whose edges partition E(K (n, p)). A holey hamiltonian decomposition is a holey 2-factorization of K (n, p) where each holey 2-factor is a connected subgraph of K (n, p) − V i for some i satisfying 1 ≤ i ≤ p. A (holey) k-factorization of K (n, p) is said to be fair if the edges between each pair of parts are shared as evenly as possible among the permitted (holey) factors. In this paper the existence of fair holey hamiltonian decompositions of K (n, p) is completely settled. This result simultaneously settles the existence of cycle frames of type n p for cycles of the longest length, being a companion for results in the literature for frames with short cycle length.

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

“…Theorem 2.8. ( [7,9,11,17,18,30,35,40]) There exists a (k, 1)-CF(g u ) if and only if g ≡ 0 (mod 2), g(u − 1) ≡ 0 (mod k), u ≥ 3 when k is even, u ≥ 4 when k is odd, except a (6, 1)-CF( 63 ).…”