On the Holroyd-Talbot Conjecture for Sparse Graphs

TL;DR

This paper investigates the Holroyd-Talbot conjecture for sparse graphs, proving it holds for various classes of sparse graphs with bounds on the size of intersecting independent sets.

Contribution

It extends the conjecture to sparse graphs, establishing conditions under which the maximum intersecting family of independent sets is a star.

Findings

Proves the conjecture for graphs with bounded average degree for r ≤ O(n^{1/3})

Establishes the conjecture for bounded degree graphs for r ≤ O(n^{1/2})

Shows the conjecture holds for trees with few split vertices for r ≤ O(n^{1/2})

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 $r\le O(n^{1/2})$.