Constructing graphs with no independent transversals

TL;DR

This paper introduces a systematic method for constructing graphs with large block sizes and no independent transversals, unifying previous results and providing new extremal constructions and counterexamples.

Contribution

The authors develop a simple, unified construction method for graphs lacking independent transversals, extending known results and presenting new extremal and counterexample graphs.

Findings

Unified construction method for graphs with no IT.

Derived classical extremal constructions in a streamlined way.

Provided new minimal graphs with maximum degree two and no IT.

Abstract

Given a graph $G$ and a partition $P$ of its vertex set, an independent transversal (IT) is an independent set of $G$ that contains one vertex from each block in $P$. Various sufficient conditions for the existence of an IT have been established, and a common theme for many of them is that the block sizes are sufficiently large compared to the maximum degree of $G$. Consequently, there has been interest in constructing graphs with no IT which demonstrate that these bounds on the block sizes are best possible. We describe a simple systematic method for constructing vertex-partitioned graphs with large block sizes and no IT. Unifying previous constructions, we use our method to derive classical extremal constructions due to Jin (1992), Yuster (1997), and Szab\'o and Tardos (2006) in streamlined fashion. For our new results, we describe extremal constructions of minimal graphs with maximum degree two and no IT, generalizing a result of Aharoni, Holzman, Howard, and Spr\"ussel (2015). We construct a family of locally sparse graphs with no IT, complementing an asymptotic result of Loh and Sudakov (2007). We describe new and smaller counterexamples to a list coloring conjecture of Reed (1999), which was originally disproved by Bohman and Holzman (2002). We disprove a conjecture of Aharoni, Alon, and Berger (2016) about IT's in graphs without large induced stars. We answer negatively a question of Aharoni, Holzman, Howard, and Spr\"ussel (2015) about extremal graphs with no IT, but we also prove that a useful variation of their question does hold.