On factors of independent transversals in $k$-partite graphs

TL;DR

This paper investigates the existence of a set of disjoint independent transversals in special $k$-partite graphs, improving the known bounds and answering a longstanding question about the minimal size needed for such factors.

Contribution

The authors significantly improve the upper bound on the size needed for a factor of independent transversals in $k$-partite graphs, from $2k-2$ to approximately $1.78k$, for large $k$.

Findings

Established that $f(k) \,\le\, 1.78k$ for large $k$

Improved upon the greedy algorithm bound of $2k-2$

Provided a tighter bound answering MacKeigan's question

Abstract

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 $n_0$ such that every $[k,n,1]$-graph has a factor of independent transversals assuming $n \ge n_0$. Several known conjectures imply that for $k \ge 2$, $f(k)=k$ 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 $f(k) \le 2k-2$, 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 $f(k) \le 1.78k$ for all $k$ sufficiently large, answering a question of MacKeigan.