Chromatic index of graphs with no cycle with a unique chord

The class C of graphs that do not contain a cycle with a unique chord was recently studied by Trotignon and Vušković (in press) [23], who proved for these graphs strong structure results which led to solving the recognition and vertex-colouring problems in polynomial time. In the present paper, we investigate how these structure results can be applied to solve the edge-colouring problem in the class. We give computational complexity results for the edge-colouring problem restricted to C and to the subclass C′ composed of the graphs of C that do not have a 4-hole. We show that it is NP-complete to determine whether the chromatic index of a graph is equal to its maximum degree when the input is restricted to regular graphs of C with fixed degree Δ≥3. For the subclass C′, we establish a dichotomy: if the maximum degree is Δ=3, the edge-colouring problem is NP-complete, whereas, if Δ≠3, the only graphs for which the chromatic index exceeds the maximum degree are the odd holes and the odd order complete graphs, a characterization that solves edge-colouring problem in polynomial time. We determine two subclasses of graphs in C′ of maximum degree 3 for which edge-colouring is polynomial. Finally, we remark that a consequence of one of our proofs is that edge-colouring in NP-complete for r-regular tripartite graphs of degree Δ≥3, for r≥3.