Erdos-Ko-Rado theorems for a family of trees

A family of sets is intersecting if any two sets in the family intersect. Given a graph G and an integer r >= 1, let I-(r)(G) denote the family of independent sets of size r of G. For a vertex v of G, let I-v((r))(G) denote the family of independent sets of size r that contain v. This family is called an r-star and v is its centre. Then G is said to be r-EKR if no intersecting subfamily of I-(r)(G) is bigger than the largest r-star, and if every maximum size intersecting subfamily of I-(r)(G) is an r-star, then G is said to be strictly r-EKR. Let mu(G) denote the minimum size of a maximal independent set of G. Holroyd and Talbot conjectured that if 2r <= mu(G), then G is r-EKR, and it is strictly r-EKR if 2r < mu(G). This conjecture has been investigated for several graph classes, but not trees (except paths). In this note, we present a result for a family of trees. A depth-two claw is a tree in which every vertex other than the root has degree 1 or 2 and every vertex of degree 1 is at distance 2 from the root. We show that if G is a depth-two claw, then G is strictly r-EKR if 2r <= mu(G) + 1, confirming the conjecture of Holroyd and Talbot for this family. Hurlbert and Kamat had conjectured that one can always find a largest r-star of a tree whose centre is a leaf. Baber and Borg have independently shown this to be false. We show that, moreover, for all integers n >= 2 and d >= 3, there exists a positive integer r such that there is a tree where the centre of the largest r-star is a vertex of degree n at distance d from every leaf. (C) 2017 Elsevier B.V. All rights reserved.