T‐Cores for (Δ+t)‐edge‐coloring

Abstract

AbstractWe extend the edge‐coloring notion of core (subgraph induced by the vertices of maximum degree) to ‐core (subgraph induced by the vertices with ), and find a sufficient condition for ‐edge‐coloring. In particular, we show that for any , if the ‐core of has multiplicity at most , with its edges of multiplicity inducing a multiforest, then . This extends previous work of Ore, Fournier, and Berge and Fournier. A stronger version of our result (which replaces the multiforest condition with a vertex‐ordering condition) generalizes a theorem of Hoffman and Rodger about cores of ‐edge‐colorable simple graphs. In fact, our bounds hold not only for chromatic index, but for the fan number of a graph, a parameter introduced by Scheide and Stiebitz as an upper bound on chromatic index. We are able to give an exact characterization of the graphs such that whenever has as its ‐core.