On the Holroyd-Talbot Conjecture for Sparse Graphs

Abstract

Given a graph $G$, let $\mu(G)$ denote the size of the smallest maximal independent set in $G$. A family of subsets is called a star if some element is in every set of the family. A split vertex has degree at least 3. Holroyd and Talbot conjectured the following Erd\H{o}s-Ko-Rado type statement about intersecting families of independent sets in graphs: if $1\le r\le \mu(G)/2$ then there is an intersecting family of independent $r$-sets of maximum size that is a star. In this paper we prove similar statements for sparse graphs on $n$ vertices: roughly, for graphs of bounded average degree with $r\le O(n^{1/3})$, for graphs of bounded degree with $r\le O(n^{1/2})$, and for trees having a bounded number of split vertices with $r\le O(n^{1/2})$.