On intersecting families of independent sets in trees

TL;DR

This paper investigates intersecting families of independent sets in trees, proving the leaf property for certain classes of trees and characterizing maximum star centers, extending classical combinatorial results to graph theory.

Contribution

It establishes the leaf property for spiders and pendant trees for all relevant r, and characterizes maximum star centers in these trees, advancing understanding of intersecting independent sets.

Findings

Spiders have the leaf property for all r ≤ α(G).

Pendant trees have the leaf property for all r ≤ α(G).

Characterization of maximum star centers in these trees.

Abstract

A family of sets is intersecting if every pair of its sets intersect. A star is a family with some element (a center) in each of its sets. The classical 1961 result of Erd\H{o}s, Ko, and Rado states that every intersecting family of r-sets with $r\leq n/2$ has size at most that of a star. We say that graph G is r-EKR if, among all intersecting families of independent r-sets of G, the largest is attained by a star. In 2005 Holroyd and Talbot conjectured that every graph G is r-EKR for all $1\leq r\leq \mu(G)/2$, where $\mu(G)$ is the size of the smallest maximal independent set in G. We verified the conjecture in 2011 for all chordal graphs containing an isolated vertex. For graphs without isolated vertices it is difficult to determine the center of the largest star, which is often necessary to prove that they are EKR. A tree has the leaf property if its largest star occurs on one of its leaves. We proved that every tree T has the leaf property when $r\leq 4$, and in 2017 Borg and other authors gave examples of families of trees not having the leaf property when $r\geq 5$. A split vertex in a tree is a vertex of degree at least 3. A spider is a tree with exactly one split vertex. Here we prove that all spiders have the leaf property for all $r\leq \alpha(G)$, where $\alpha(G)$ is the independence number of $G$, and we characterize which of its leaves are maximum star centers. A pendant tree is one for which each of its split vertices is adjacent to a leaf. Here we show that all pendant trees have the leaf property for all $r\leq \alpha(G)$. We also consider pendant trees with exactly two split vertices and provide partial results on the locations of their maximum star centers.