On acyclic b-chromatic number of cubic graphs

TL;DR

This work investigates the acyclic b-chromatic number $A_b(G)$ for cubic graphs, i.e., the maximum number of colors in an acyclic coloring that cannot be further reduced by acyclic recolorings. Building on the recoloring framework and known bounds, it leverages results of Jakovac and Klavžar and constructive colorings on families such as generalized Petersen graphs and $(0,j)$-prisms. The main findings show that all cubic graphs except the prism $K_2\Box K_3$ satisfy $A_b(G)\in\{4,5\}$, with infinitely many achieving $A_b(G)=4$; generalized Petersen graphs exhibit both 4 and 5 cases depending on parameters, and prisms and their generalizations can force $A_b$ up to 5 under size conditions. The paper also provides infinite families with $A_b(G)=4$, and outlines open questions on complexity, snarks, and conjectures linking girth to acyclic colorability, pointing to directions for future work.

Abstract

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.

Figures (11)

• Figure 1: An acyclic b-coloring of a cubic graph. Black vertices are acyclic b-vertices: the one colored with $1$ is of type A and those colored with $2,3,4$ and $5$ are of type B.
• Figure 2: Petersen graph, prism $K_2\Box K_3$, $K_{3,3}$ and $G_1$, and their acyclic b-colorings (acyclic b-vertices are black).
• Figure 3: Corrected b-colorings of JaKl: first graph of the second line of Figure 14 and first, fourth, and fifth graphs from Figure 15; that are now acyclic. On the last graph, we have corrected a minor coloring error in the third graph from Figure 13 from JaKl. The original colors are indicated in brackets, and black vertices represent b-vertices.
• Figure 4: Graph $H_3$ and the construction of a graph $C(T)$ from a cubic tree $T\cong K_{1,3}$.
• Figure 5: Graphs $G(6,2)$ and $G(7,3)$.

Theorems & Definitions (10)

• Theorem 3.1: JaKl
• Theorem 3.2
• Theorem 3.3
• Lemma 3.4
• Theorem 3.5
• Corollary 3.6
• Theorem 4.1
• Theorem 4.2
• Theorem 5.1
• Conjecture 6.4