Linear colorings of subcubic graphs

TL;DR

This paper proves that all connected subcubic graphs, except C5 and K3,3, can be linearly colored from lists of size four with a polynomial-time algorithm, confirming a recent conjecture.

Contribution

It confirms a conjecture on list linear coloring of subcubic graphs and provides a constructive, linear-time algorithm for such colorings.

Findings

Valid linear coloring exists for all connected subcubic graphs except C5 and K3,3.

The proof is constructive and yields a linear-time coloring algorithm.

The result extends understanding of list colorings in graphs with maximum degree three.

Abstract

A linear coloring of a graph is a proper coloring of the vertices of the graph so that each pair of color classes induce a union of disjoint paths. In this paper, we prove that for every connected graph with maximum degree at most three and every assignment of lists of size four to the vertices of the graph, there exists a linear coloring such that the color of each vertex belongs to the list assigned to that vertex and the neighbors of every degree-two vertex receive different colors, unless the graph is $C_5$ or $K_{3,3}$. This confirms a conjecture raised by Esperet, Montassier, and Raspaud. Our proof is constructive and yields a linear-time algorithm to find such a coloring.