Rainbow independent sets on dense graph classes

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 $\varphi :I\to \mathcal{I}$ where for each $v\in I$, $v$ is contained in $\varphi (v)$. 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.