On edge-colouring indifference graphs

Abstract

Vizing's theorem states that the chromatic index χ′( G) of a graph G is either the maximum degree Δ( G) or Δ( G) + 1. A graph G is called overfull if ¦E(G)¦> Δ(G)∥¦V(G)¦/2∥. A sufficient condition for χ′( G) = Δ( G) + 1 is that G contains an overfull subgraph H with Δ( H) = Δ( 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 χ′ = Δ.