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

Abstract

A unichord in a graph is an edge that is the unique chord of a cycle. A square is an induced cycle on four vertices. A graph is 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.

Edge-colouring and total-colouring chordless graphs

“…Our proof uses a different strategy based on the existence of an extreme decomposition tree, in which one of the decomposition blocks is 2-sparse. This leads to our third and main motivation for this work: to understand how such kind of decomposition results, which are classically applied to the design of vertex colouring algorithms, can be useful in the development of algorithms for other colouring problems, in particular edge-colouring [24] and total-colouring [25,26] -the present work is successful in the sense that our chosen class of chordless graphs showed to be fruitful for the development of polynomial-time edge-colouring and total-colouring algorithms. Section 2 reviews the decomposition result for chordless graphs established in [21].…”