A Survey of the Holroyd-Talbot Conjecture

TL;DR

This paper surveys the 20-year history of the Holroyd-Talbot Conjecture, which generalizes the Erdős-Ko-Rado Theorem to intersecting families of independent sets in graphs, highlighting key supporting results.

Contribution

It provides a comprehensive overview of research progress and supporting evidence for the Holroyd-Talbot Conjecture over two decades.

Findings

Supported the conjecture for various classes of graphs

Identified conditions under which the conjecture holds

Summarized partial results and open problems

Abstract

A family of sets is intersecting if every pair of its members has an element in common. Such a family of sets is called a star if some element is in every set of the family. Given a graph $G$, let $\mu(G)$ denote the size of the smallest maximal independent set of $G$. In 2005, Holroyd and Talbot conjectured the following generalization of the Erd\H{o}s-Ko-Rado Theorem: for $1\le r\le \mu(G)/2$, there is a maximum size intersecting family of independent $r$-sets that is a star. In this paper we present the history of this conjecture and survey the results that have supported it over the last 20 years.