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arXiv · 1309.2749

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

Raphael C. S. Machado·Celina M. H. de Figueiredo·Nicolas Trotignon
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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.

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Raphael C. S. Machado, Celina M. H. de Figueiredo, Nicolas Trotignon. 2013-09-11. Complexity of colouring problems restricted to unichord-free and \{square,unichord\}-free graphs. https://doi.org/10.1016/j.dam.2012.02.016

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