摘要
图G的一个圆环r-染色(r≥2)是将G的每个顶点v对应到一个周长为r的圆上的点的一个映射f,使得对于G中任意的边xy,f(x)和f(y)在圆上的距离不小于1.G的圆环色数χc(G)是G存在圆环r-染色的最小实数r.符号图的圆环染色和图的圆环染色基本相同,不同的是对于负边xy,我们要求f(x)和f(y)的对点在圆上的距离不小于1.符号图(G,σ)的圆环色数是使得(G,σ)在圆环r-染色的最小实数r.本文证明:对于任意正整数k和实数ε>0,存在整数g使得对于任意树宽至多为k的符号图(G,σ),如果(G,-σ)的负围长至少是g,那么(G,σ)的圆环染色数至多是2+ε.
A circular r-coloring of a graph G is a mapping f from V(G)to the points of a circle of circumference r such that for every xy∈E(G),f(x)and f(y)are at distance at least1.The smallest r for which G admits a circular r-coloring is the circular chromatic number of G.A circular r-coloring of a signed graph(G,σ)is the same as a circular r-coloring of a graph except that for every negative edge xy,f(x)and the antipodal of f(y)are at distance at least1.The smallest r for which(G,σ)admits a circular r-coloring is the circular chromatic number of(G,σ).This paper proves the following result:For any positive integer k and real numberε>0,there is an integer g such that for any signed graph(G,σ)with treewidth at most k,if(G,-σ)has negative girth at least g,then(G,σ)has circular chromatic number at most 2+ε.
作者
周欢
朱绪鼎
ZHOU Huan;ZHU Xuding(School of Mathematical Sciences,Zhejiang Normal University,Jinhua,Zhejiang,321004,P.R.China)
出处
《数学进展》
CSCD
北大核心
2023年第5期795-803,共9页
Advances in Mathematics（China）
基金
Supported by NSFC(Nos.11971438,U20A2068)
关键词
符号图
环染色
树宽
奇围长
负围长
signed graph
circular coloring
treewidth
odd girth
negative girth
作者简介
朱绪鼎,E-mail:xdzhu@zjnu.edu.cn
