The Erdős-Ko-Rado properties of various graphs containing singletons

Let G = ( V , E ) be a graph. For r ≥ 1 , let I G ( r ) be the family of independent vertex r -sets of G . For v ∈ V ( G ) , let I G ( r ) ( v ) denote the star { A ∈ I G ( r ) : v ∈ A } . G is said to be r-EKR if there exists v ∈ V ( G ) such that | A | ≤ | I G ( r ) ( v ) | for any non-star family A of pair-wise intersecting sets in I G ( r ) . If the inequality is strict, then G is strictly r -EKR. 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 2 r 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 2 r is no larger than μ ( G ) , the size of a smallest maximal independent vertex set of G , then G is r -EKR, and strictly so if 2 r < μ ( G ) . We show that in fact, if G ∈ Γ and 2 r is no larger than the independence number of G , then G is r -EKR; we do this by proving the result for all graphs that are in a suitable larger set Γ ′ ⊋ Γ . We also confirm the conjecture for graphs in an even larger set Γ ″ ⊋ Γ ′ .