Very Well-Covered Graphs with the Erdős-Ko-Rado Property

Abstract

A family of independent $r$-sets of a graph $G$ is an $r$-star if every set in the family contains some fixed vertex $v$. A graph is $r$-EKR if the maximum size of an intersecting family of independent $r$-sets is the size of an $r$-star. Holroyd and Talbot conjecture that a graph is $r$-EKR as long as $1\le r\le \frac{\mu (G)}{2}$, where $\mu (G)$ is the minimum size of a maximal independent set. It is suspected that the smallest counterexample to this conjecture is a well-covered graph. Here we consider the class of very well-covered graphs $G^{\ast }$ obtained by appending a single pendant edge to each vertex of $G$. We prove that the pendant complete graph $K_n^{\ast }$ is $r$-EKR when $n\ge 2r$ and strictly so when $n>2r$. Pendant path graphs $P_n^{\ast }$ are also explored and the vertex whose $r$-star is of maximum size is determined.

10 pages