Erd\"os-Ko-Rado theorems for chordal and bipartite graphs

TL;DR

This paper extends Erd"os-Ko-Rado theorems to specific classes of graphs, establishing conditions under which the graphs exhibit the r-EKR property, including chordal graphs, chains of complete graphs, ladder graphs, and trees.

Contribution

It proves new Erd"os-Ko-Rado results for disjoint unions of chordal graphs, chains of complete graphs, and provides preliminary results for ladder graphs and trees.

Findings

Disjoint union of chordal graphs with a singleton is r-EKR if r ≤ mu(G)/2.

Erd"os-Ko-Rado results established for chains of complete graphs.

Preliminary results obtained for ladder graphs and trees.

Abstract

One of the more recent generalizations of the Erd\"os-Ko-Rado theorem, formulated by Holroyd, Spencer and Talbot, defines the Erd\"os-Ko-Rado property for graphs in the following manner: for a graph G and a positive integer r, G is said to be r-EKR if no intersecting subfamily of the family of all independent vertex sets of size r is larger than the largest star, where a star centered at a vertex v is the family of all independent sets of size $r$ containing v. In this paper, we prove that if G is a disjoint union of chordal graphs, including at least one singleton, then G is r-EKR if $r\leq mu(G)/2$, where mu(G) is the minimum size of a maximal independent set. We will also prove Erd\"os-Ko-Rado results for chains of complete graphs, which are a class of chordal graphs obtained by blowing up edges of a path into complete graphs. We also consider similar problems for ladder graphs and trees, and prove preliminary results for these graphs.