Extending edge colorings of distance-3 matchings in the Cartesian product of graphs

TL;DR

The authors study extending precolored distance-3 matchings in Cartesian products of Class 1 graphs to a proper edge coloring using at most $\sum_{i=1}^k Δ(G_i)$ colors. They develop a constructive framework based on a canonical coloring, brick-neighborhoods, and local square rotations to adjust precolored edges without breaking color validity, proving a general extension theorem and a specialized result for odd-cycle products. These results imply the Casselgren et al. conjecture in broad settings and yield corollaries for products with $K_2$ and bipartite factors, while outlining key open questions for broader degree configurations and mixed parity products. The work advances understanding of edge-coloring extension problems in graph products and informs algorithmic approaches for coloring product graphs.

Abstract

We investigate the problem of extending partial edge colorings in Cartesian products of graphs, with a particular focus on cases where the precolored edges form a matching. Casselgren, Granholm, and Petros conjectured that any precolored distance-3 matching in $G = C^d_{2k}$ can be extended to a $2d$-edge coloring. In this paper, we prove a theorem that implies this conjecture. Especially, our main result establishes that a precolored distance-3 matching in the Cartesian product of certain class 1 graphs can be extended to an edge coloring that uses at most as many colors as the chromatic index, provided that certain degree conditions are satisfied. In the second part of the paper, we extend these results to Cartesian products of other types of graphs as well.

Theorems & Definitions (5)

• proof
• proof
• proof : Proof of Theorem \ref{['general']}
• proof
• proof : Proof of Theorem \ref{['odd-times-odd']}