By a $z$-coloring of a graph $G$ we mean any proper vertex coloring consisting of the color classes $C_1, \ldots, C_k$ such that $(i)$ for any two colors $i$ and $j$ with $1 \leq i < j \leq k$, any vertex of color $j$ is adjacent to a vertex of color $i$, $(ii)$ there exists a set $\{u_1, \ldots, u_k\}$ of vertices of $G$ such that $u_j \in C_j$ for any $j \in \{1, \ldots, k\}$ and $u_k$ is adjacent to $u_j$ for each $1 \leq j \leq k$ with $j \not=k$, and $(iii)$ for each $i$ and $j$ with $i \not= j$, the vertex $u_j$ has a neighbor in $C_i$. Denote by $z(G)$ the maximum number of colors used in any $z$-coloring of $G$. Denote the Grundy and {\rm b}-chromatic number of $G$ by $\Gamma(G)$ and ${\rm b}(G)$, respectively. The $z$-coloring is an improvement over both the Grundy and b-coloring of graphs. We prove that $z(G)$ is much better than $\min\{\Gamma(G), {\rm b}(G)\}$ for infinitely many graphs $G$ by obtaining an infinite sequence $\{G_n\}_{n=3}^{\infty}$ of graphs such that $z(G_n)=n$ but $\Gamma(G_n)={\rm b}(G_n)=2n-1$ for each $n\geq 3$. We show that acyclic graphs are $z$-monotonic and $z$-continuous. Then it is proved that to decide whether $z(G)=\Delta(G)+1$ is $NP$-complete even for bipartite graphs $G$. We finally prove that to recognize graphs $G$ satisfying $z(G)=\chi(G)$ is $coNP$-complete, improving a previous result for the Grundy number.

翻译：图的 $z$-着色是指一种顶点真着色，其色类为 $C_1, \ldots, C_k$，满足以下条件：(i) 对于任意两个颜色 $i$ 和 $j$（$1 \leq i < j \leq k$），每个颜色 $j$ 的顶点都与某个颜色 $i$ 的顶点相邻；(ii) 存在一组顶点 $\{u_1, \ldots, u_k\}$，使得对于每个 $j \in \{1, \ldots, k\}$，有 $u_j \in C_j$，且对于每个 $1 \leq j \leq k$ 且 $j \not= k$，$u_k$ 与 $u_j$ 相邻；(iii) 对于每一对不同的 $i$ 和 $j$，顶点 $u_j$ 在 $C_i$ 中有一个邻点。记 $z(G)$ 为图 $G$ 在任何 $z$-着色中使用的最大颜色数。分别记 $\Gamma(G)$ 和 ${\rm b}(G)$ 为图 $G$ 的 Grundy 数和 b-色数。$z$-着色是对图的 Grundy 着色和 b-着色的改进。我们通过构造一个无限图序列 $\{G_n\}_{n=3}^{\infty}$，使得对于每个 $n \geq 3$，有 $z(G_n)=n$ 但 $\Gamma(G_n)={\rm b}(G_n)=2n-1$，证明了对于无穷多个图 $G$，$z(G)$ 远优于 $\min\{\Gamma(G), {\rm b}(G)\}$。我们证明无环图是 $z$-单调且 $z$-连续的。进而证明，即使对于二分图 $G$，判定 $z(G)=\Delta(G)+1$ 也是 $NP$-完全的。最后，我们证明识别满足 $z(G)=\chi(G)$ 的图 $G$ 是 $coNP$-完全的，改进了此前关于 Grundy 数的结果。