Locally irregular edge-coloring of subcubic graphs

TL;DR

This paper proves that certain classes of subcubic graphs, including claw-free, cycle permutation, and generalized Petersen graphs, can be edge-colored with at most 3 colors to achieve local irregularity, advancing understanding of decomposable graphs.

Contribution

It establishes that several classes of decomposable subcubic graphs admit a 3-color locally irregular edge-coloring, reducing the known upper bound for these graphs.

Findings

Claw-free graphs with maximum degree 3 are 3-colorable.

Cycle permutation graphs are 3-colorable.

Generalized Petersen graphs are 3-colorable.

Abstract

A graph is {\em locally irregular} if no two adjacent vertices have the same degree. A {\em locally irregular edge-coloring} of a graph $G$ is such an (improper) edge-coloring that the edges of any fixed color induce a locally irregular graph. Among the graphs admitting a locally irregular edge-coloring, i.e., {\em decomposable graphs}, only one is known to require $4$ colors, while for all the others it is believed that $3$ colors suffice. In this paper, we prove that decomposable claw-free graphs with maximum degree $3$, all cycle permutation graphs, and all generalized Petersen graphs admit a locally irregular edge-coloring with at most $3$ colors. We also discuss when $2$ colors suffice for a locally irregular edge-coloring of cubic graphs and present an infinite family of cubic graphs of girth $4$ which require $3$ colors.