Rainbow independent sets on dense graph classes

TL;DR

This paper investigates the existence of rainbow independent sets in dense graph classes, expanding known results to include graphs with bounded neighborhood diversity and r-powers of bounded expansion graphs.

Contribution

It introduces two new dense graph classes that satisfy the rainbow independent set property, extending previous work on this combinatorial property.

Findings

Graphs of bounded neighborhood diversity satisfy the property.

r-powers of graphs in bounded expansion classes satisfy the property.

The results broaden understanding of rainbow independent sets in dense graphs.

Abstract

Given a family $\mathcal{I}$ of independent sets in a graph, a rainbow independent set is an independent set $I$ such that there is an injection $\phi\colon I\to \mathcal{I}$ where for each $v\in I$, $v$ is contained in $\phi(v)$. Aharoni, Briggs, J. Kim, and M. Kim [Rainbow independent sets in certain classes of graphs. arXiv:1909.13143] determined for various graph classes $\mathcal{C}$ whether $\mathcal{C}$ satisfies a property that for every $n$, there exists $N=N(\mathcal{C},n)$ such that every family of $N$ independent sets of size $n$ in a graph in $\mathcal{C}$ contains a rainbow independent set of size $n$. In this paper, we add two dense graph classes satisfying this property, namely, the class of graphs of bounded neighborhood diversity and the class of $r$-powers of graphs in a bounded expansion class.