Let $G$ be a multigraph and $L\,:\,E(G) \to 2^\mathbb{N}$ be a list assignment on the edges of $G$. Suppose additionally, for every vertex $x$, the edges incident to $x$ have at least $f(x)$ colors in common. We consider a variant of local edge-colorings wherein the color received by an edge $e$ must be contained in $L(e)$. The locality appears in the function $f$, i.e., $f(x)$ is some function of the local structure of $x$ in $G$. Such a notion is a natural generalization of traditional local edge-coloring. Our main results include sufficient conditions on the function $f$ to construct such colorings. As corollaries, we obtain local analogs of Vizing and Shannon's theorems, recovering a recent result of Conley, Greb\'ik and Pikhurko.

翻译：设$G$为一个多重图，$L\,:\,E(G) \to 2^\mathbb{N}$为$G$边上的一个列表分配。进一步假设，对于每个顶点$x$，与$x$相关联的边至少有$f(x)$种共同颜色。我们考虑局部边着色的一种变体，其中边$e$接收的颜色必须包含在$L(e)$中。局部性体现在函数$f$上，即$f(x)$是$x$在$G$中局部结构的某个函数。这一概念是传统局部边着色的自然推广。我们的主要结果包括构造此类着色的函数$f$的充分条件。作为推论，我们得到了Vizing定理和Shannon定理的局部版本，并重现了Conley、Grebík和Pikhurko最近的一个结果。