arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 0911.4199math.COcs.DM
收藏

图的2-色色数的独立数上界

Upper bounds for the 2-hued chromatic number of graphs in terms of the independence number

Arash Ahadi, Ali Dehghan

备注：
Dynamic chromatic number; conditional (k, 2)-coloring; 2-hued chromatic number; 2-hued coloring; Independence number; Probabilistic method
发表信息：
Discrete Applied Mathematics 160(2012)2142--2146
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

图$G$的双色着色（也称为条件$(k, 2)$着色和动态着色）是这样的着色，使得对于度至少为$2$的每个顶点$v\in V(G)$，$v$的邻居至少接收$2$颜色。使$G$具有$ k $颜色的2色调着色的最小整数$k$被称为$G$的{ \it 2色调色数}，并由$χ_2(G)$表示。本文证明了：若$G$是正则图，则有$ χ_{2}(G)- χ(G) \leq 2 \log _{2}(α(G)) +\mathcal{O}(1) $;若$G$是图且有$δ(G)\geq 2$，则有$ χ_{2}(G)- χ(G) \leq 1+\lceil \sqrt[δ-1]{4Δ^{2}} \rceil ( 1+ \log _{\frac{2Δ(G)}{2Δ(G)-δ(G)}} (α(G)) ) $;若$G$是图，则有$ χ_{2}(G)- χ(G) \leq 2+ \min \lbrace α^{\prime}(G),\frac{α(G)+ω(G)}{2}\rbrace $ .

英文摘要：

A 2-hued coloring of a graph $G$ (also known as conditional $(k, 2)$-coloring and dynamic coloring) is a coloring such that for every vertex $v\in V(G)$ of degree at least $2$, the neighbors of $v$ receive at least $2$ colors. The smallest integer $k$ such that $G$ has a 2-hued coloring with $ k $ colors, is called the {\it 2-hued chromatic number} of $G$ and denoted by $χ_2(G)$. In this paper, we will show that if $G$ is a regular graph, then $ χ_{2}(G)- χ(G) \leq 2 \log _{2}(α(G)) +\mathcal{O}(1) $ and if $G$ is a graph and $δ(G)\geq 2$, then $ χ_{2}(G)- χ(G) \leq 1+\lceil \sqrt[δ-1]{4Δ^{2}} \rceil ( 1+ \log _{\frac{2Δ(G)}{2Δ(G)-δ(G)}} (α(G)) ) $ and in general case if $G$ is a graph, then $ χ_{2}(G)- χ(G) \leq 2+ \min \lbrace α^{\prime}(G),\frac{α(G)+ω(G)}{2}\rbrace $.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑