More results on the $z$chromatic number of graphs
Abstract
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 bcoloring 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 $\Gamma(G_n)={\rm b}(G_n)=2n1$ 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2023
 DOI:
 10.48550/arXiv.2302.01306
 arXiv:
 arXiv:2302.01306
 Bibcode:
 2023arXiv230201306K
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics
 EPrint:
 Submitted To Disc. Appl. Math. on September 8, 2022