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 $d\left(v\right)+\mu \left(v\right)>\mathrm{\Delta }+t$), 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 $\chi \prime \left(G\right)\le \mathrm{\Delta }+t$. 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 $\text{Fan}\left(G\right)\le \mathrm{\Delta }\left(G\right)+t$ whenever $G$ has $H$ as its $t$‐core.

中文翻译：

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用于（Δ+ t）边缘着色的t型芯

我们将核的边缘着色概念（由最大度数的顶点所引起的子图）扩展为$Ť$‐核心（由顶点导出的子图$v$ 与 $d（v）+\mu （v）>\mathrm{\Delta }+Ť$），并为 $（\mathrm{\Delta }+Ť）$边缘着色。特别是，我们证明了对于任何$Ť\ge 0$，如果 $Ť$的核心 $G$ 最多有多重性 $Ť+\mathrm{1个}$，具有多重优势 $Ť+\mathrm{1个}$ 诱导多林，然后 $\chi \prime （G）\le \mathrm{\Delta }+Ť$。这扩展了Ore，Fournier和Berge和Fournier的先前工作。我们结果的更强版本（用顶点排序条件代替了多林条件）概括了Hoffman和Rodger关于定理的定理。$\mathrm{\Delta }$边缘可着色的简单图形。实际上，我们的界限不仅适用于色度指数，还适用于图形的扇数，这是Scheide和Stiebitz引入的参数，作为色度指数的上限。我们能够给出图的精确表征$H$ 这样 $\text{风扇}（G）\le \mathrm{\Delta }（G）+Ť$ 每当 $G$ 具有 $H$ 就像它一样 $Ť$-核心。