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On a conjecture concerning the property of chromatic polynomials with negative variable

Yan Yang

Abstract

Let $G$ be a graph of order $n$ and $P(G,x)$ be the chromatic polynomial of $G$. Dong, Ge, Gong, Ning, Ouyang, and Tay (J. Graph Theory 96(2021) 343) conjectured that $\frac{d^k}{dx^k} \bigl( \ln[(-1)^n P(G, x)] \bigr) < 0$ holds for all $k \geq 2$ and $x \in (-\infty, 0)$. We prove this conjecture for all $k \geq 2 $ and $ x\leq -10Δk $, in which $Δ$ is the maximum degree of $G$.

On a conjecture concerning the property of chromatic polynomials with negative variable

Abstract

Let be a graph of order and be the chromatic polynomial of . Dong, Ge, Gong, Ning, Ouyang, and Tay (J. Graph Theory 96(2021) 343) conjectured that holds for all and . We prove this conjecture for all and , in which is the maximum degree of .
Paper Structure (2 sections, 8 theorems, 32 equations)

This paper contains 2 sections, 8 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Main results

Key Result

Theorem 1.1

Let $G$ be a graph of order $n$ and $\eta: E\to \{1, 2, \dots, |E|\}$ be a bijection. Then, where $a_i(G)$ is the number of spanning subgraphs of $G$ with $n - i$ edges which do not contain broken cycles.

Theorems & Definitions (13)

  • Theorem 1.1: W1932
  • Theorem 1.2: S01
  • Conjecture 1.3: D21
  • Lemma 1.4
  • proof
  • Lemma 2.1: F15
  • Theorem 2.2: F15
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 3 more