The Open University ’ s repository of research publications and other research outputs Graphs with the Erdos

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 (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 (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 2μ(G), then G is r-EKR, and if r < 1 2μ(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. MSC: 05C35, 05D05