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Let $G$ be a multigraph and $L\,:\,E(G) \to 2^\mathbb{N}$ be a listassignment on the edges of $G$. Suppose additionally, for every vertex $x$, theedges incident to $x$ have at least $f(x)$ colors in common. We consider avariant of local edge-colorings wherein the color received by an edge $e$ mustbe 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 anatural generalization of traditional local edge-coloring. Our main resultsinclude 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.

Multigraph edge-coloring with local list sizes