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Article Dans Une Revue Discrete Applied Mathematics Année : 2014

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

Résumé

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.

Dates et versions

hal-01230809 , version 1 (19-11-2015)

Identifiants

Citer

Raphael C.S. Machado, Celina M.H. de Figueiredo, Nicolas Trotignon. Complexity of colouring problems restricted to unichord-free and { square,unichord }-free graphs. Discrete Applied Mathematics, 2014, 164, ⟨10.1016/j.dam.2012.02.016⟩. ⟨hal-01230809⟩
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