Complexity of colouring problems restricted to unichord-free and \{square,unichord\}-free graphs

TL;DR

This paper proves that, except for K4, square-free unichord-free graphs with maximum degree 3 can be total-colored with four colors, and provides a polynomial-time algorithm for this coloring, contrasting with NP-complete edge-coloring.

Contribution

It establishes a polynomial-time total-coloring method for a specific class of graphs, clarifying the complexity of coloring problems within this class.

Findings

Total-coloring with four colors is possible for all such graphs except K4.

The proof leads to a polynomial-time coloring algorithm.

Edge-coloring remains NP-complete for this class.

Abstract

A \emph{unichord} in a graph is an edge that is the unique chord of a cycle. A \emph{square} is an induced cycle on four vertices. A graph is \emph{unichord-free} if none of its edges is a unichord. We give a slight restatement of a known structure theorem for unichord-free graphs and use it to show that, with the only exception of the complete graph $K_4$, every square-free, unichord-free graph of maximum degree~3 can be total-coloured with four colours. Our proof can be turned into a polynomial time algorithm that actually outputs the colouring. This settles the class of square-free, unichord-free graphs as a class for which edge-colouring is NP-complete but total-colouring is polynomial.