高围长图的${\rm b}^*$-着色与$z$-着色

中文摘要 AI 辅助

在图$G$的正常顶点着色$c$中，若顶点$u$与每个其他颜色类中的一个顶点相邻，则称$u$为$b$-顶点。${\rm b}^*$-着色是一种正常着色，其中每个$b$-顶点都与每个其他颜色类中的一个$b$-顶点相邻。Grundy着色是通过首次适应（贪心）着色过程得到的正常着色。图$G$的$z$-着色是同时为${\rm b}^*$-着色和Grundy着色的着色。${\rm b}^*$-色数（记为${\rm b}^*(G)$）是图$G$的${\rm b}^*$-着色中使用的最大颜色数，$z$-色数（记为$z(G)$）同理。所有图都可使用多项式时间着色启发式方法得到${\rm b}^*$-着色和$z$-着色。设${\rm m}^*(G)$为满足以下条件的最大整数$k$：图$G$中度数至少为$k$的顶点有$k$个度数至少为$k$的邻居。本文运用列表着色技术证明，若图$G$的围长至少为7，则${\rm b}^*(G)={\rm m}^*(G)+1$；当${\rm m}^*=3$时，围长至少为6的图也可得到类似结果；最后证明，若图的围长至少为$2k+2$且包含某特定树作为普通子图，则$z(G)\geq k$。

英文摘要

In a proper vertex coloring $c$ of a graph $G$, a vertex $u$ is called a b-vertex if $u$ is adjacent to a vertex in every other color class. A ${\rm b}^{\ast}$-coloring is a proper coloring in which a b-vertex is adjacent to a b-vertex in every other color class. A Grundy coloring is a proper coloring obtained by the First-Fit (greedy) coloring procedure. A $z$-coloring of $G$ is a ${\rm b}^{\ast}$-coloring that is also a Grundy coloring. The ${\rm b}^{\ast}$-chromatic number (resp., $z$-chromatic number), denoted by ${\rm b}^{\ast}(G)$ (resp., $z(G)$), is the maximum number of colors used in a ${\rm b}^{\ast}$-coloring (resp., $z$-coloring) of $G$. Every graph admits a ${\rm b}^{\ast}$-coloring and a $z$-coloring that can be found using a polynomial-time coloring heuristic. Let ${\rm m}^{\ast}(G)$ be the largest integer $k$ such that a vertex of degree at least $k$ in $G$ has $k$ neighbors of degree at least $k$. We employ list-coloring techniques to prove that if $G$ has a girth of at least $7$, then ${\rm b}^{\ast}(G) = {\rm m}^{\ast}(G)+ 1$. A similar result is obtained for graphs of girth at least $6$ when ${\rm m}^{\ast}=3$. Finally, we obtain some results for the $z$-chromatic number. We prove that if the girth is at least $2m^{\ast}(G)+4$ and $G$ contains a specific tree as an ordinary subgraph, then $z(G)= m^{\ast}(G)+1$.

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