More results on the $z$-chromatic number of graphs
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909126223724544 |
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| author | Khaleghi, Abbas Zaker, Manouchehr |
| author_facet | Khaleghi, Abbas Zaker, Manouchehr |
| contents | 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 $Γ(G)$ 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 $\min\{Γ(G), {\rm b}(G)\}$ for infinitely many graphs $G$ by obtaining an infinite sequence $\{G_n\}_{n=3}^{\infty}$ of graphs such that $z(G_n)=n$ but $Γ(G_n)={\rm b}(G_n)=2n-1$ for each $n\geq 3$. We show that acyclic graphs are $z$-monotonic and $z$-continuous. Then it is proved that to decide whether $z(G)=Δ(G)+1$ is $NP$-complete even for bipartite graphs $G$. We finally prove that to recognize graphs $G$ satisfying $z(G)=χ(G)$ is $coNP$-complete, improving a previous result for the Grundy number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_01306 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | More results on the $z$-chromatic number of graphs Khaleghi, Abbas Zaker, Manouchehr Combinatorics Discrete Mathematics 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 $Γ(G)$ 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 $\min\{Γ(G), {\rm b}(G)\}$ for infinitely many graphs $G$ by obtaining an infinite sequence $\{G_n\}_{n=3}^{\infty}$ of graphs such that $z(G_n)=n$ but $Γ(G_n)={\rm b}(G_n)=2n-1$ for each $n\geq 3$. We show that acyclic graphs are $z$-monotonic and $z$-continuous. Then it is proved that to decide whether $z(G)=Δ(G)+1$ is $NP$-complete even for bipartite graphs $G$. We finally prove that to recognize graphs $G$ satisfying $z(G)=χ(G)$ is $coNP$-complete, improving a previous result for the Grundy number. |
| title | More results on the $z$-chromatic number of graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2302.01306 |