Graphs with the Erdős–Ko–Rado property

Abstract

For a graph G vertex v of G and integer r ⩾ 1 , we denote the family of independent r-sets of V ( G ) by I ( r ) ( G ) and the subfamily { A ∈ I ( r ) ( G ) : v ∈ A } by I v ( r ) ( G ) ; such a subfamily is called a star. Then, G is said to be r-EKR if no intersecting subfamily of I ( r ) ( G ) is larger than the largest star in I ( r ) ( G ) . If every intersecting subfamily of I v ( r ) ( G ) of maximum size is a star, then G is said to be strictly r- EKR. We show that if a graph G is r -EKR then its lexicographic product with any complete graph is r-EKR. For any graph G , we define μ ( G ) to be the minimum size of a maximal independent vertex set. We conjecture that, if 1 ⩽ r ⩽ 1 2 μ ( G ) , then G is r-EKR, and if r < 1 2 μ ( G ) , then G is strictly r-EKR. This is known to be true when G is an empty graph, a cycle, a path or the disjoint union of complete graphs. We show that it is also true when G is the disjoint union of a pair of complete multipartite graphs.