Stars on trees

TL;DR

This paper disproves a conjecture about the maximum number of independent sets containing a specific vertex in trees, showing counterexamples for larger values of r where the conjecture fails.

Contribution

The authors construct explicit counterexamples to the conjecture for r ≥ 5, demonstrating the conjecture's limitations in certain cases.

Findings

Counterexamples for r ≥ 5 in trees.

The conjecture does not hold universally for all r.

Largest r with non-empty independent sets at x is 2k+1.

Abstract

For a positive integer $r$ and a vertex $v$ of a graph $G$, let $\mathcal{I}_G^{(r)}(v)$ denote the set of all independent sets of $G$ that have exactly $r$ elements and contain $v$. Hurlbert and Kamat conjectured that for any $r$ and any tree $T$, there exists a leaf $z$ of $T$ such that $|\mathcal{I}_T^{(r)}(v)| \leq |\mathcal{I}_T^{(r)}(z)|$ for each vertex $v$ of $T$. They proved the conjecture for $r \leq 4$. For any $k \geq 3$, we construct a tree $T_k$ that has a vertex $x$ such that $x$ is not a leaf of $T_k$, $|\mathcal{I}_{T_k}^{(r)}(z)| < |\mathcal{I}_{T_k}^{(r)}(x)|$ for any leaf $z$ of $T_k$ and any $5 \leq r \leq 2k+1$, and $2k+1$ is the largest integer $s$ for which $\mathcal{I}_{T_k}^{(s)}(x)$ is non-empty. Therefore, the conjecture is not true for $r \geq 5$.