Erdös-Ko-Rado theorems for chordal and bipartite graphs

One of the more recent generalizations of the Erdös-Ko-Rado theorem, formulated by Holroyd, Spencer and Talbot [10], de nes the Erdös-Ko-Rado property for graphs in the following manner: for a graph G, vertex v ∈ G and some integer r ≥ 1, denote the family of independent r-sets of V (G) by J (r)(G) and the subfamily {A ∈ J (r)(G) : v ∈ A} by J (r) v (G), called a star. Then, G is said to be r-EKR if no intersecting subfamily of J (r)(G) is larger than the largest star in J (r)(G). In this paper, we prove that if G is a disjoint union of chordal graphs, including at least one singleton, then G is r-EKR if r ≤ μ(G) 2 , where μ(G) is the minimum size of a maximal independent set. We will also prove Erdös-Ko-Rado results for chains of complete graphs, which are a class of chordal graphs obtained by blowing up edges of a path into complete graphs. We also consider similar problems for ladder graphs and trees, and prove preliminary results for these graphs.