On z-coloring and b∗-coloring of graphs as improved variants of the b-coloring

Abstract

Let $G$ be a simple graph and $c$ a proper vertex coloring of $G$. A vertex $u$ is called b-vertex in $(G,c)$ if all colors except $c\left(u\right)$ appear in the neighborhood of $u$. By a $b^{\ast }$-coloring of $G$ using colors $\{1,\dots ,k\}$ we define a proper vertex coloring $c$ such that there is a b-vertex $u$ (called nice vertex) such that $u$ is adjacent to a b-vertex of color $j$ for each $j\in \{1,\dots ,k\}$ with $j\ne c\left(u\right)$. The $b^{\ast }$-chromatic number of $G$, denoted by $b^{\ast }\left(G\right)$ is the largest integer $k$ such that $G$ has a $b^{\ast }$-coloring using $k$ colors. Every graph $G$ admits a $b^{\ast }$-coloring which is an improvement over the famous b-coloring. A z-coloring of $G$ is a coloring $c$ using colors $\{1,2,\dots ,k\}$ containing a nice vertex of color $k$ such that for each two colors $i<j$, each vertex of color $j$ has a neighbor of color $i$ in the graph (i.e. $c$ is obtained from a Grundy-coloring of $G$). We prove that $b^{\ast }\left(G\right)$ cannot be approximated within any constant factor unless $P=\mathrm{NP}$. We obtain results for $b^{\ast }$-coloring and z-coloring of block graphs, cacti, $P_4$-sparse graphs, pseudo-split graphs and graphs with girth greater than 4. A linear 0-1 programming model is also presented for z-coloring of graphs. The positive results suggest that researches can be focused on $b^{\ast }$-coloring (or z-coloring) instead of b-coloring of graphs.