Multigraph edge-coloring with local list sizes

Let G be a multigraph and $L:E\left(G\right)\to 2^N$ be a list assignment for the edges of G. Suppose additionally, for every vertex x, the edges incident to x have at least $f\left(x\right)$ colors in common. We consider a variant of local edge-colorings wherein the color received by an edge e must be contained in $L\left(e\right)$. The locality appears in the function f, i.e., $f\left(x\right)$ 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, Grebík, and Pikhurko.