On cross-intersecting families of independent sets in graphs

TL;DR

This paper extends Hilton's cross-intersection theorem to graph theory, establishing bounds for cross-intersecting independent sets in various graphs, including chordal graphs and cycles, with a proven conjecture for cycles.

Contribution

It formulates a graph-theoretic analogue of Hilton's theorem, extending results to chordal graphs and cycles, and proposes a conjecture for all graphs.

Findings

Proved bounds for uniform cross-intersecting independent sets in complete graphs.

Extended the theorem to chordal graphs.

Confirmed the conjecture for cycles on n vertices.

Abstract

Let A_1,...,A_k be a collection of families of subsets of an n-element set. We say that this collection is cross-intersecting if for any i,j in [k] with i not equal to j, A in A_i and B in A_j implies that the intersection of A and B is nonempty. We consider a theorem of Hilton which gives a best possible upper bound on the sum of the cardinalities of uniform cross-intersecting subfamilies. We formulate a graph-theoretic analogue of Hilton's cross-intersection theorem, similar to the one developed by Holroyd, Spencer and Talbot for the Erdos-Ko-Rado theorem. In particular we build on a result of Borg and Leader for signed sets and prove a theorem for uniform cross-intersecting subfamilies of independent vertex subsets of a disjoint union of complete graphs. We proceed to obtain a result for a much larger class of graphs, namely chordal graphs and propose a conjecture for all graphs. We end by proving this conjecture for the cycle on n vertices.