Independence Number and Maximal Chromatic Polynomials of Connected Graphs

IF 0.6 4区数学 Q3 MATHEMATICS

Graphs and Combinatorics Pub Date : 2024-08-07 DOI:10.1007/s00373-024-02824-2

Shude Long, Junliang Cai

引用次数: 0

Abstract

Let ${\mathcal {C}}_{k}(n)$ denote the family of all connected graphs of order n with chromatic number k. In this paper we show that the conjecture proposed by Tomescu which if $x\ge k\ge 4$ and $G\in {\mathcal {C}}_{k}(n)$, then

$$\begin{aligned} P(G,x)\le (x)_{k} (x-1)^{n-k} \end{aligned}$$

holds under the additional condition that G has an independent cut-set T of size at most 2 such that the number of components in $G{\setminus } T$ is equal to the independence number of G.

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连通图的独立数和最大色度多项式

让 ${\mathcal {C}}_{k}(n)$ 表示色度数为 k 的 n 阶所有连通图的族。本文将证明 Tomescu 提出的猜想，即如果 $x\ge k\ge 4$ and$G\in {\mathcal {C}}_{k}(n)$, then $$\begin{aligned}P(G,x)\le (x)_{k}(x-1)^{n-k}\end{aligned}$$holds under the additional condition that G has an independent cut-set T of size at most 2 such that the number of components in $G{\setminus } T$ is equal to the independence number of G.

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Graphs and Combinatorics 数学-数学

CiteScore

1.00

自引率

14.30%

发文量

160

审稿时长

6 months

期刊介绍： Graphs and Combinatorics is an international journal devoted to research concerning all aspects of combinatorial mathematics. In addition to original research papers, the journal also features survey articles from authors invited by the editorial board.