The Erd®s-Ko-Rado property of various graphs containing singletons

Let G = (V,E) be a graph. Let IG be the family of all independent sets of G. For r ≥ 1, let I G := {I ∈ IG : |I| = r}. For v ∈ V (G), let I (r) G (v) denote the star {A ∈ I G : v ∈ A}. G is said to be (strictly) r-EKR if there exists v ∈ V (G) such that (|A| < |I G (v)|) |A| ≤ |I (r) G (v)| for any non-star family A of pair-wise intersecting sets in I G . Let Γ be the family of graphs that are disjoint unions of complete graphs, paths, cycles, including at least one singleton. Holroyd, Spencer and Talbot proved that if G ∈ Γ and 2r is no larger than the number of connected components of G, then G is r-EKR. However, Holroyd and Talbot conjectured that if G is any graph and 2r ≤ μ(G) := min{|I| : I ∈ IG, I maximal}, then G is r-EKR, and strictly so if 2r < μ(G). We show that in fact G is r-EKR if 2r ≤ α(G) := max{|I| : I ∈ IG}; we do this by proving the result for all graphs that are in a suitable larger set Γ′ ) Γ. We also con rm the conjecture for graphs in an even larger set Γ′′ ) Γ′.