Total chromatic number of unichord-free graphs

A unichord   is an edge that is the unique chord of a cycle in a graph. The class CC of unichord-free graphs — that is, graphs that do not contain, as an induced subgraph, a cycle with a unique chord — was recently studied by Trotignon and Vušković (2010) [24], who proved strong structure results for these graphs and used these results to solve the recognition and vertex-colouring problems. Machado et al. (2010) [18] determined the complexity of the edge-colouring problem in the class CC and in the subclass C′C′ obtained from CC by forbidding squares. In the present work, we prove that the total-colouring problem is NP-complete when restricted to graphs in CC. For the subclass C′C′, we establish the validity of the Total Colouring Conjecture by proving that every non-complete {square, unichord}-free graph of maximum degree at least 4 is Type 1.