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

Daniel Iľkovič, Jun Yan

TL;DR

The paper addresses leaf-maximising behavior in the HK generalization of EKR for independent sets. It proves that perfect $r$-ary trees are HK by constructing injections from any center $v$ to a leaf $ ext{l}$, via a decomposition into $ ext{A}_{v, ext{l}}$, $ ext{B}_{v, ext{l}}$, and $ ext{C}_{v, ext{l}}$, and two parity-based injection schemes built on the CAS gadget. The CAS operation provides a robust, injective way to transfer local configurations across isomorphic subtrees while preserving independence. The results extend to forests of perfect trees, yielding a leaf-based maximisation rule determined by arity and height parity, and an open question remains about which $k$-EKR thresholds hold for perfect $r$-ary trees. The techniques blend combinatorial injections with a programmable swap mechanism to resolve conflicts across symmetric tree structures.

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 Erdős-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 $|\mathcal{I}_{\mathcal{T}}^k(v)|$.

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

TL;DR

The paper addresses leaf-maximising behavior in the HK generalization of EKR for independent sets. It proves that perfect -ary trees are HK by constructing injections from any center to a leaf , via a decomposition into , , and , and two parity-based injection schemes built on the CAS gadget. The CAS operation provides a robust, injective way to transfer local configurations across isomorphic subtrees while preserving independence. The results extend to forests of perfect trees, yielding a leaf-based maximisation rule determined by arity and height parity, and an open question remains about which -EKR thresholds hold for perfect -ary trees. The techniques blend combinatorial injections with a programmable swap mechanism to resolve conflicts across symmetric tree structures.

Abstract

Given a graph , the family of all independent sets of size containing a fixed vertex is called a star with centre , and is denoted by . Motivated by a generalisation of the Erdős-Ko-Rado Theorem to the setting of independent sets in graphs, Hurlbert and Kamat conjectured that for every tree and every , the maximum of can always be attained by a leaf of . 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 -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 that is the disjoint union of perfect trees with possibly varying sizes and arities, and determine the leaf that maximises .
Paper Structure (6 sections, 7 theorems, 1 equation, 1 figure)

This paper contains 6 sections, 7 theorems, 1 equation, 1 figure.

Table of Contents

  1. Introduction
  2. Perfect $r$-ary trees
  3. The Conditional Alternating Swap (CAS)
  4. Case 1: $d(v,\ell)$ is even
  5. Case 2: Odd Distance $d(v,\ell)$
  6. Multiple disjoint perfect trees

Key Result

Theorem 1.5

Let $T$ be a perfect $r$-ary tree. For any integer $k \geq 1$, any vertex $v \in V(T)$, and any leaf $\ell \in V(T)$, we have $|\mathcal{I}_{T}^{k}(v)| \leq |\mathcal{I}_{T}^{k}(\ell)|$.

Figures (1)

  • Figure 1: Example run of CAS with $r=2$. Vertices in $I_i$ are black, and vertices to be modified in each step are circled in red.

Theorems & Definitions (18)

  • Definition 1.1
  • Definition 1.2
  • Conjecture 1.3: HK
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1: Conditional Alternating Swap
  • Lemma 2.2: Injectivity of CAS
  • proof
  • Lemma 2.3
  • ...and 8 more