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arXiv · 0903.4203

Erdös-Ko-Rado theorems for chordal and bipartite graphs

Glenn Hurlbert·Vikram Kamat
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Abstract

One of the more recent generalizations of the Erdös-Ko-Rado theorem, formulated by Holroyd, Spencer and Talbot, defines the Erdös-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 Erdös-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.

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Glenn Hurlbert, Vikram Kamat. 2009-07-15. Erdös-Ko-Rado theorems for chordal and bipartite graphs. https://arxiv.org/abs/0903.4203

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