Total chromatic number of unichord-free graphs

Abstract A unichord is an edge that is the unique chord of a cycle in a graph. The class C 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 Vuskovic (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 C and in the subclass C ′ obtained from C by forbidding squares. In the present work, we prove that the total-colouring problem is NP-complete when restricted to graphs in C . For the subclass 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.

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