  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 $t\geq 0$, if
the $t$-core of $G$ has multiplicity at most $t+1$, with its edges of
multiplicity $t+1$ inducing a multiforest, then $\chi'(G) \leq \Delta+t$. 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-colourable simple graphs. In fact, our bounds hold not
only for chromatic index, but for the \emph{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
$\mathrm{Fan}(G) \leq \Delta(G)+t$ whenever $G$ has $H$ as its $t$-core.
