Fair 1-Factorizations and Fair Holey 1-Factorizations of Complete Multipartite Graphs

A k-factor of a graph 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 V1, ..., Vp are the p parts of V (K(n, p)), then a holey k-factor of deficiency Vi of K(n, p) is a k-factor of K(n, p)− Vi for some i satisfying 1 ≤ i ≤ p. Hence a holey k-factorization is a set of holey k-factors whose edges partition E(K(n, p)). A (holey) k-factorization of K(n, p) is said to be fair if between each pair of parts the color classes have size within one of each other (so the edges are shared “evenly” among the permitted (holey) factors). In this talk a sketch of a proof will be given for the existence of (fair 1-factorizations of K(n, p) and) fair holey 1-factorizations of K(n, p). Fair holey 1-factorizations of K(n, p) can be used to provide a new construction for symmetric quasigroups of order np with holes of size n. Such quasigroups have the additional property that the permitted symbols are shared as evenly as possible among the cells in each n× n “box”. These quasigroups are in some sense as far from frames produced by direct products as possible.