On acyclic b-chromatic number of cubic graphs
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| Format: | Preprint |
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2025
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| _version_ | 1866917056556826624 |
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| author | Anholcer, Marcin Cichacz, Sylwia Peterin, Iztok |
| author_facet | Anholcer, Marcin Cichacz, Sylwia Peterin, Iztok |
| contents | Let $G$ be a graph. An acyclic $k$-coloring of $G$ is a map $c:V(G)\rightarrow \{1,\dots,k\}$ such that $c(u)\neq c(v)$ for any $uv\in E(G)$ and the subgraph induced by the vertices of any two colors $i,j\in \{1,\dots,k\}$ is a forest. If every vertex $v$ of a color class $V_i$ misses a color $\ell_v\in\{1,\dots,k\}$ in its closed neighborhood, then every $v\in V_i$ can be recolored with $\ell_v$ and we obtain a $(k-1)$-coloring of $G$. If a new coloring $c'$ is also acyclic, then such a recoloring is an acyclic recoloring step and $c'$ is in relation $\triangleleft_a$ with $c$. The acyclic b-chromatic number $A_b(G)$ of $G$ is the maximum number of colors in an acyclic coloring where no acyclic recoloring step is possible. Equivalently, it is the maximum number of colors in a minimum element of the transitive closure of $\triangleleft_a$. In this paper, we consider $A_b(G)$ of cubic graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_01532 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On acyclic b-chromatic number of cubic graphs Anholcer, Marcin Cichacz, Sylwia Peterin, Iztok Combinatorics Discrete Mathematics 05C15 Let $G$ be a graph. An acyclic $k$-coloring of $G$ is a map $c:V(G)\rightarrow \{1,\dots,k\}$ such that $c(u)\neq c(v)$ for any $uv\in E(G)$ and the subgraph induced by the vertices of any two colors $i,j\in \{1,\dots,k\}$ is a forest. If every vertex $v$ of a color class $V_i$ misses a color $\ell_v\in\{1,\dots,k\}$ in its closed neighborhood, then every $v\in V_i$ can be recolored with $\ell_v$ and we obtain a $(k-1)$-coloring of $G$. If a new coloring $c'$ is also acyclic, then such a recoloring is an acyclic recoloring step and $c'$ is in relation $\triangleleft_a$ with $c$. The acyclic b-chromatic number $A_b(G)$ of $G$ is the maximum number of colors in an acyclic coloring where no acyclic recoloring step is possible. Equivalently, it is the maximum number of colors in a minimum element of the transitive closure of $\triangleleft_a$. In this paper, we consider $A_b(G)$ of cubic graphs. |
| title | On acyclic b-chromatic number of cubic graphs |
| topic | Combinatorics Discrete Mathematics 05C15 |
| url | https://arxiv.org/abs/2511.01532 |