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- W2962808480 abstract "Abstract 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 μ ( G ) denote the minimum size of a maximal independent set of G . Holroyd and Talbot conjectured that if 2 r ≤ μ ( G ) , then G is r -EKR, and it is strictly r -EKR if 2 r μ ( 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 2 r ≤ μ ( 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." @default.
- W2962808480 created "2019-07-30" @default.
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- W2962808480 date "2015-06-26" @default.
- W2962808480 modified "2023-09-27" @default.
- W2962808480 title "Erdős-Ko-Rado Theorems for a Family of Trees" @default.
- W2962808480 hasPublicationYear "2015" @default.
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