On Edge-Colouring Indifference Graphs

. Vizing's theorem states that the chromatic index Ø 0 (G) of a graph G is either the maximum degree \Delta(G) or \Delta(G) + 1. A graph G is called overfull if jE(G)j ? \Delta(G)bjV (G)j=2c. A sufficient condition for Ø 0 (G) = \Delta(G) + 1 is that G contains an overfull subgraph H with \Delta(H ) = \Delta(G). Plantholt proved that this condition is necessary for graphs with a universal vertex. In this paper, we conjecture that, for indifference graphs, this is also true. As supporting evidence, we prove this conjecture for general graphs with three maximal cliques and with no universal vertex, and for indifference graphs with odd maximum degree. For the latter subclass, we prove that Ø 0 = \Delta. 1 Introduction An edge-colouring of a graph is an assignment of colours to its edges such that no adjacent edges have the same colour. The chromatic index of a graph is the minimum number of colours required to produce an edge-colouring for that graph. In this paper, we address..