t-Cores for ( Δ + 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 ), and find a sufficient condition for $\left(\mathrm{\Delta }+t\right)$-edge-coloring. In particular, we show that for any $t\ge 0$, if the $t$-core of $G$ has multiplicity at most $t+1$, with its edges of multiplicity $t+1$ inducing a multiforest, then $\chi \prime \text{◂≤▸}\left(G\right)\le \mathrm{\Delta }+t$. 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 $\mathrm{\Delta }$-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 $\text{◂...▸}\text{Fan}\left(G\right)\le \text{◂+▸}\mathrm{\Delta }\left(G\right)+t$ whenever $G$ has $H$ as its $t$-core.