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On the zero-free region for the chromatic polynomial of claw-free graphs with and without induced square and induced diamond

Paula M. S. Fialho, Aldo Procacci

TL;DR

This work extends zero-free region results for chromatic polynomials to claw-free graphs by introducing the pair independence ratio $\kappa$ and analyzing two hereditary classes ${\cal G}_0$ and ${\cal G}_1$. Using a Whitney broken circuit/partition-scheme approach and the Penrose partition scheme, the authors express $P_{\mathbb{G}}(q)$ via Penrose forests and derive explicit bounds on a Penrose-forest generating function, yielding a disk $|q|< C^i_{\kappa} Δ$ where $C^i_{\kappa}$ grows with $\kappa$; a sharper bound is obtained for ${\cal G}_1$ (square-free and diamond-free). The core proof combines inductive control of the Penrose expansion with combinatorial bounds on independent pairs in neighborhoods (via Mantel-type arguments) to establish nonvanishing in a radius that translates to a zero-free region for $P_{\mathbb{G}}(q)$. The results improve on previous bounds by Bencs and Regts and provide computable constants $C^i_{\kappa}$, enhancing understanding of chromatic roots for claw-free graphs and related subfamilies.

Abstract

Given a claw-free graph $G=(V,E)$ with maximum degree $Δ$, we define the parameter $κ\in [0,1]$ as $κ={\max_{v\in V}|I_v|\over \lfloorΔ^2/4\rfloor}$ where $I_v$ is the set of all independent pairs in the neighborhood of $v$. We refer to $κ$ as the pair independence ratio of $G$. We prove that for any claw-free graph $G$ with pair independence ratio at most $κ$ the zeros of its chromatic polynomial $P_G(q)$ lie inside the disk $D=\{q\in \mathbb{C}:~|q|< C_κ^0Δ\}$, where $C_κ^0$ is an increasing function of $κ\in [0,1]$. If $G$ is also square-free and diamond free, the function $C_κ^0$ can be replaced by a sharper function $C_κ^1$. These bounds constitute an improvement upon results recently given by Bencs and Regts in ''Improved bounds on the zeros of the chromatic polynomial of graphs and claw-free graphs''.

On the zero-free region for the chromatic polynomial of claw-free graphs with and without induced square and induced diamond

TL;DR

This work extends zero-free region results for chromatic polynomials to claw-free graphs by introducing the pair independence ratio and analyzing two hereditary classes and . Using a Whitney broken circuit/partition-scheme approach and the Penrose partition scheme, the authors express via Penrose forests and derive explicit bounds on a Penrose-forest generating function, yielding a disk where grows with ; a sharper bound is obtained for (square-free and diamond-free). The core proof combines inductive control of the Penrose expansion with combinatorial bounds on independent pairs in neighborhoods (via Mantel-type arguments) to establish nonvanishing in a radius that translates to a zero-free region for . The results improve on previous bounds by Bencs and Regts and provide computable constants , enhancing understanding of chromatic roots for claw-free graphs and related subfamilies.

Abstract

Given a claw-free graph with maximum degree , we define the parameter as where is the set of all independent pairs in the neighborhood of . We refer to as the pair independence ratio of . We prove that for any claw-free graph with pair independence ratio at most the zeros of its chromatic polynomial lie inside the disk , where is an increasing function of . If is also square-free and diamond free, the function can be replaced by a sharper function . These bounds constitute an improvement upon results recently given by Bencs and Regts in ''Improved bounds on the zeros of the chromatic polynomial of graphs and claw-free graphs''.
Paper Structure (7 sections, 8 theorems, 63 equations, 1 table)

This paper contains 7 sections, 8 theorems, 63 equations, 1 table.

Table of Contents

  1. Introduction and results
  2. Preliminaries
  3. Trees, forests and partition schemes in a graph
  4. A bound on $T_{{\mathbb{G}},v}(y)$ when ${\mathbb{G}}$ is a claw-free graph
  5. Proof of Theorem \ref{['teopri']}
  6. A key proposition and an auxiliary lemma
  7. Proof of Proposition \ref{['pro1']}

Key Result

Theorem 1.2

Let ${\mathbb{G}}$ be a graph with maximum degree at most $\Delta\ge 3$ and pair independence parameter $\kappa$. If ${\mathbb{G}}\in {\cal G}_i$ with $i=0,1$, then there exists a constant $C^i_{\kappa}$ such that $P_{\mathbb{G}}(q) \neq 0$ for any $q \in \mathbb{C}$ satisfying where constant $C^i_{\kappa}$ increases with $\kappa$ and ranges from $C^1_{0}\le 3$ to $C^0_{1}\le 3.81$.

Theorems & Definitions (11)

  • Definition 1.1
  • Theorem 1.2
  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3: Penrose partition scheme
  • Lemma 2.4
  • Corollary 2.5
  • Proposition 3.1
  • Lemma 3.2
  • Proposition 4.1
  • ...and 1 more