Graphs of low average degree without independent transversals

TL;DR

This paper demonstrates the optimality of a known bound for the existence of independent transversals in graphs with low average degree, providing constructions that show the bound cannot be improved.

Contribution

The authors construct examples of forests with partitions and low average degree where independent transversals do not exist, proving the tightness of previous results and methods.

Findings

Constructed forests with no independent transversal at near-threshold degrees.

Showed that entropy compression methods are tight for this problem.

Extended results to hypergraph variants.

Abstract

An independent transversal of a graph $G$ with a vertex partition $\mathcal P$ is an independent set of $G$ intersecting each block of $\mathcal P$ in a single vertex. Wanless and Wood proved that if each block of $\mathcal P$ has size at least $t$ and the average degree of vertices in each block is at most $t/4$, then an independent transversal of $\mathcal P$ exists. We present a construction showing that this result is optimal: for any $\varepsilon > 0$ and sufficiently large $t$, there is a family of forests with vertex partitions whose block size is at least $t$, average degree of vertices in each block is at most $(\frac14+\varepsilon)t$, and there is no independent transversal. This unexpectedly shows that methods related to entropy compression such as the Rosenfeld-Wanless-Wood scheme or the Local Cut Lemma are tight for this problem. Further constructions are given for variants of the problem, including the hypergraph version.