Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $t$-cores for $(\Delta+t)$-edge-colouring   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.

10
The categories of this paper are: q-bio
24
The categories of this paper are: dg-ga
27
The categories of this paper are: comp-gas
23
The categories of this paper are: solv-int
14
The categories of this paper are: math-ph
12
The categories of this paper are: nucl-th
8
The categories of this paper are: hep-ex
16
The categories of this paper are: stat
33
The categories of this paper are: plasm-ph
35
The categories of this paper are: mtrl-th
7
The categories of this paper are: hep-th
26
The categories of this paper are: bayes-an
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
6
The categories of this paper are: astro-ph
29
The categories of this paper are: funct-an