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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 slightrestatement of a known structure theorem for unichord-free graphs and use it toshow that, with the only exception of the complete graph $K_4$, everysquare-free, unichord-free graph of maximum degree~3 can be total-coloured withfour colours. Our proof can be turned into a polynomial time algorithm thatactually outputs the colouring. This settles the class of square-free,unichord-free graphs as a class for which edge-colouring is NP-complete buttotal-colouring is polynomial.

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