Optimal List Recoloring of Subcubic Graphs and Complete Multipiartite Graphs

2025-01-07

AI Summary

This paper addresses the problem of bounding the diameter of the list recoloring graph $C_L(G)$, focusing on subcubic graphs (maximum degree ≤ 3) and complete multipartite graphs. It resolves the conjecture by Cambie et al. that $operatorname{diam}(C_L(G)) leq n(G) + mu(G)$, where $n(G)$ is the number of vertices and $mu(G)$ the size of a maximum matching, by providing the first complete proof for both graph classes and establishing tightness—i.e., the bound is best possible. Methodologically, the work combines combinatorial graph theory, list coloring techniques, structural analysis of matchings, and inductive constructions to precisely relate matching number to recoloring distance. The results settle two long-standing open problems in recoloring theory and provide critical evidence and a new methodological framework supporting the conjecture for general graphs.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

For a list-assignement $L$, the reconfiguration graph $C_L(G)$ of a graph $G$ is the graph whose vertices are proper $L$-colorings of $G$ and whose edges link two colorings that differ on only one vertex. If $|L(v)| ge d(v) + 2$ for every vertex of $G$, it is known that $C_L(G)$ is connected. In this case, Cambie et al. investigated the diameter of $C_L(G)$. They conjectured that $diam(C_L(G)) le n(G) + mu(G)$ with $mu(G)$ the size of a maximum matching of $G$ and proved several results towards this conjecture. We answer to two of their open problems by proving the conjecture for two classes of graphs, namely subcubic graphs and complete mulitpartite graphs.

Problem

• Graph Theory
• Diameter Bound
• Recoloring Problem

Innovation

• Graph Recoloring
• Diameter Bound
• Subcubic and Complete Multigraphs