t-Cores for ( plus t)edge-coloring

We extend the edge-coloring notion of core (subgraph induced by the vertices of maximum degree) to t-core (subgraph induced by the vertices v with d(v) + mu(v) > Delta + t ), and find a sufficient condition for (Delta + t)-edge-coloring. In particular, we show that for any t >= 0, if the t-core of G has multiplicity at most t + 1, with its edges of multiplicity t + 1 inducing a multiforest, then chi'(G) <= Delta + t. 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 Delta-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 H such that Fan(G) <= Delta(G) + t whenever G has H as its t-core.