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

Factoring Graphs

This chapter focuses on Hall's Theorem, introduced by British mathematician Philip Hall, and its connection to graph theory. It first considers problems that ask whether some collection of objects can be matched in some way to another collection of objects, with particular emphasis on how different types of schedulings are possible using a graph. It then examines one popular version of Hall's work, a statement known as the Marriage Theorem, the occurrence of matchings in bipartite graphs, Tutte's Theorem, Petersen's Theorem, and the Petersen graph. Peter Christian Julius Petersen introduced the Petersen graph to show that a cubic bridgeless graph need not be 1-factorable. The chapter concludes with an analysis of 1-factorable graphs, the 1-Factorization Conjecture, and 2-factorable graphs.

A generalization of Hall’s theorem

Let [Formula: see text] be a finite group, whose order has [Formula: see text] prime divisors. In this paper, we prove that if [Formula: see text] has a [Formula: see text]-complement for [Formula: see text] prime divisors [Formula: see text] of [Formula: see text] and [Formula: see text] has no section isomorphic to [Formula: see text]. Then [Formula: see text] is solvable, which generalizes a theorem of Hall.