Total coloring of (sub)cubic Halin graphs

TL;DR

This paper proves that for cubic and subcubic Halin graphs, only finitely many require five colors for total coloring, advancing understanding of the total coloring conjecture in this class.

Contribution

It completely characterizes total coloring of cubic and subcubic Halin graphs, showing finitely many need five colors, thus resolving a specific case of the total coloring problem.

Findings

Finitely many cubic Halin graphs require five colors

Total coloring conjecture holds for all but finitely many such graphs

Complete classification of total coloring for these graphs

Abstract

Total coloring of a graph is a coloring of its vertices and edges such that adjacent or incident elements receive distinct colors. Total coloring conjecture (stipulating that the total chromatic number of a graph $G$ is at most $\Delta(G)+2$) is known to be true for subcubic graphs -- five colors are always enough. However, deciding whether a total coloring with only four colors exists remains a difficult problem, even in the class of bipartite cubic graphs. We solve the problem completely for cubic and subcubic Halin graphs, proving that there are only finitely many such graphs requiring five colors.