A Graph-theoretical Generalization of Berge’s Analogue of the Erdős-Ko-Rado Theorem

Abstract

A family A of r-subsets of the vertex set V(G) of a graph G is intersecting if any two of the r-subsets have a non-empty intersection. The graph G is r-EKR if a largest intersecting family A of independent r-subsets of V(G) may be obtained by taking all independent r-subsets containing some particular vertex.

In this paper, we show that if G consists of one path P raised to the power κ ₀ ≥ 1, and s cycles ₁ C, ₂ C, ..., _s C raised to the powers κ ₁, κ ₂, ..., κ _s respectively, with

$$ {\text{min}}\left( {\omega {\text{(}}_{\text{1}} C^{k_{\text{1}} } {\text{), }}\omega {\text{(}}_{\text{2}} C^{k_{\text{2}} } {\text{),}}...{\text{,}}\omega {\text{(}}_{\text{s}} C^{k_s } {\text{)}}} \right) \geqslant \omega {\text{(}}P^{k_{\text{0}} } {\text{)}} \geqslant {\text{2}} $$

where ω(H) denotes the clique number of H, and if G has an independent r-set (so r is not too large), then G is r-EKR. An intersecting family of the largest possible size may be found by taking all independent r-subsets of V(G) containing one of the end-vertices of the path.