Multigraph edge-coloring with local list sizes

TL;DR

This paper studies a generalized local edge-coloring problem in multigraphs with list constraints, providing new sufficient conditions and local analogs of classical theorems like Vizing and Shannon.

Contribution

It introduces a novel local edge-coloring framework with list constraints and establishes sufficient conditions for such colorings, extending classical theorems to this setting.

Findings

Derived local analogs of Vizing's theorem

Established local Shannon-type bounds

Recovered recent results of Conley, Grebík, and Pikhurko

Abstract

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.