Distribution of independent sets in perfect $r$-ary trees

TL;DR

This paper proves that in perfect r-ary trees, the largest independent set star is centered at a leaf, extending known results and constructing injections to compare stars centered at different vertices.

Contribution

It establishes the conjecture for perfect r-ary trees by constructing injections between stars centered at arbitrary vertices and leaves, and extends the result to forests of such trees.

Findings

Stars centered at leaves are largest in perfect r-ary trees.

The conjecture holds for forests of perfect trees with varying sizes and arities.

Identifies the leaf that maximizes the size of the independent set star.

Abstract

Given a graph $G$, the family of all independent sets of size $k$ containing a fixed vertex $v$ is called a star with centre $v$, and is denoted by $\mathcal{I}_G^k(v)$. Motivated by a generalisation of the Erd\H{o}s-Ko-Rado Theorem to the setting of independent sets in graphs, Hurlbert and Kamat conjectured that for every tree $T$ and every $k$, the maximum of $|\mathcal{I}_T^k(v)|$ can always be attained by a leaf of $T$. While this conjecture turns out to be false in general, it is known to hold for specific families of trees like spiders and caterpillars. In this paper, we prove that this conjecture holds for a new family of trees, the perfect $r$-ary trees, by constructing injections from stars centred at arbitrary vertices to stars centred at leaves. We also show that the analogous property holds for every forest $\mathcal{T}$ that is the disjoint union of perfect trees with possibly varying sizes and arities, and determine the leaf that maximises $|\mathcal{I}_{\mathcal{T}}^k(v)|$.