Erdős–Ko–Rado theorems for chordal graphs and trees

One of the more recent generalizations of the Erdős–Ko–Rado theorem, formulated by Holroyd, Spencer and Talbot, defines the Erdős–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 , 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 , where μ(G) is the minimum size of a maximal independent set.We also prove Erdős–Ko–Rado results for chains of complete graphs, which are special chordal graphs obtained by blowing up edges of a path into complete graphs, as well as prove preliminary results for trees.