On acyclic b-chromatic number of cubic graphs

2025-11-03

AI Summary

This paper investigates the acyclic b-chromatic number of cubic graphs—the maximum number of colors attainable in an acyclic coloring that admits no acyclic recoloring reducing the number of colors. We introduce and systematically develop this parameter, establishing its existence and extremal theory for the first time. Our approach constructs a novel combinatorial framework based on a partial order of colorings and deficiency analysis within closed neighborhoods, integrating techniques from acyclic coloring, forest-subgraph constraints, and neighborhood-structure characterization. We determine tight upper and lower bounds for the acyclic b-chromatic number of arbitrary cubic graphs and exactly compute it for several fundamental classes—including 3-regular bipartite graphs and the Petersen graph. These results extend the theory of dynamic graph colorings and provide a new paradigm for higher-order structural coloring problems.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Applications and emergent phenomena
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Let $G$ be a graph. An acyclic $k$-coloring of $G$ is a map $c:V(G) ightarrow {1,dots,k}$ such that $c(u) eq c(v)$ for any $uvin E(G)$ and the subgraph induced by the vertices of any two colors $i,jin {1,dots,k}$ is a forest. If every vertex $v$ of a color class $V_i$ misses a color $ell_vin{1,dots,k}$ in its closed neighborhood, then every $vin V_i$ 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 $ riangleleft_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 $ riangleleft_a$. In this paper, we consider $A_b(G)$ of cubic graphs.

Problem

• Determining acyclic b-chromatic number for cubic graphs
• Studying maximum colors in stable acyclic graph colorings
• Analyzing irreducibility under acyclic recoloring operations in graphs

Innovation

• Acyclic coloring with maximum color retention
• Recoloring steps preserving acyclic property
• Focus on cubic graphs structure analysis