On factors of independent transversals in k-partite graphs

A [k, n, 1]-graph is a k-partite graph with parts of order n such that the bipartite graph induced by any pair of parts is a matching. An independent transversal in such a graph is an independent set that intersects each part in a single vertex. A factor of independent transversals is a set of n pairwise-disjoint independent transversals. Let f(k) be the smallest integer n₀ such that every [k, n, 1]-graph has a factor of independent transversals assuming n ≥ n₀ . Several known conjectures imply that for k ≥ 2, f(k) = k if k is even and f(k) = k + 1 if k is odd. While a simple greedy algorithm based on iterating Hall’s Theorem shows that f(k) ≤ 2k − 2, no better bound is known and in fact, there are instances showing that the bound 2k − 2 is tight for the greedy algorithm. Here we significantly improve upon the greedy algorithm bound and prove that f(k) ≤ 1.78k for all k sufficiently large, answering a question of MacKeigan.