Bipartite 2‐Factorizations of Complete Multipartite Graphs

Abstract

It is shown that if K is any regular complete multipartite graph of even degree, and F is any bipartite 2-factor of K, then there exists a factorisation of K into F ; except that there is no factorisation of K 6,6 into F when F is the union of two disjoint 6-cycles.

Orthogonally Resolvable Cycle Decompositions

“…In particular, this implies that an OCD(K n [w], nw) exists whenever nw ≥ 3 and either w is even or n is odd. The existence of a wn-decomposition of K n [w] − I when n is even and w is odd is a direct corollary of the proof of Lemma 8 of [10]. Since Hamiltonian decompositions are always orthogonally resolvable, we have the following result.…”

Uniformly resolvable decompositions of Kv into paths on two, three and four vertices

“…A variant of the Oberwolfach problem is known as Hamilton-Waterloo problem and asks for a 2-factorization of K v having an assigned number r of factors all isomorphic to an assigned 2-regular graph H and the remaining (v − 2r − 1)/2 factors all isomorphic to another assigned 2-regular graph W . The Oberwolfach problem has been solved for a single cycle size [3] and recently for the case where the factors contain exactly two cycles [35]; for solutions to the analogous Oberwolfach Problem for complete multipartite graphs see [6,7,23,24], and [28]. As regards the Hamilton-Waterloo problem, several cases have been solved in [2,11,12], and [18], where the authors have focused on the case where both H and W are uniform, i.e., the case where all cycles of H have an assigned length h and all cycles of W have an assigned length w; in particular, the case where h = 3 and w = 4 is solved in [11], with some exceptions which are solved in [5]; the case where h = 3 and w = v is studied in [12] and [18]; finally, many other solutions to the Oberwolfach problem and to the Hamilton-Waterloo problem can be found in the literature (see, for instance, [8] and [9]).…”

A Generalization of the Hamilton–Waterloo Problem on Complete Equipartite Graphs

“…In the nonuniform case. Bryant et al [9] completely solved the case when the 2-factor is bipartite. For the Hamilton-Waterloo problem most of the results are uniform, see, for example, [3] or [11].…”

“…Theorem 1.1. [11] If m, n, and t are odd integers with n ≥ m ≥ 3, then there is a decomposition of K mnt into s C m -factors and r C n -factors if and only if s, r ≥ 0 and s + r = (mnt − 1)/2, except possibly when: r t > 1 and r = 1 or 3, or (m, n, r) = (5,9,5), (5,9,7), (7,9,5), (7,9,7), (3,13,5); r t = 1 and r ∈ {1, 2, . .…”