Constructing Graphs with No Independent Transversals

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 Szabó 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 Sprüssel (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 Sprüssel (2015) about extremal graphs with no IT, but we also prove that a useful variation of their question does hold.