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Multigraph edge-coloring with local list sizes

Abhishek Dhawan

TL;DR

The work studies local list-edge-coloring of finite multigraphs under a locality constraint $f(x,L)=\bigcap_{e\in E_G(x)} L(e)$, and proves three sufficient conditions guaranteeing a proper $L$-edge-coloring: (i) $|f(x,L)| \ge \lfloor 3\deg(x)/2\rfloor$, (ii) $|f(x,L)| \ge \deg(x)+\mu(x)$, and (iii) bipartite graphs with $|f(x,L)| \ge \deg(x)$. A chain-based augmentation framework is developed, using augmenting/path-like constructions (Shannon chains and Vizing fans) together with a potential function to drive progress toward a complete coloring, yielding polynomial-time algorithms when the local sizes are poly-bounded. These results specialize to local analogs of Shannon's and Vizing's bounds and unify prior locality results, recovering Conley--Grebik--Pikhurko-type colorings for multigraphs. The framework provides a natural, locality-aware extension of classical edge-coloring theory, with practical coloring algorithms under locality constraints.

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, Grebík and Pikhurko.

Multigraph edge-coloring with local list sizes

TL;DR

The work studies local list-edge-coloring of finite multigraphs under a locality constraint , and proves three sufficient conditions guaranteeing a proper -edge-coloring: (i) , (ii) , and (iii) bipartite graphs with . A chain-based augmentation framework is developed, using augmenting/path-like constructions (Shannon chains and Vizing fans) together with a potential function to drive progress toward a complete coloring, yielding polynomial-time algorithms when the local sizes are poly-bounded. These results specialize to local analogs of Shannon's and Vizing's bounds and unify prior locality results, recovering Conley--Grebik--Pikhurko-type colorings for multigraphs. The framework provides a natural, locality-aware extension of classical edge-coloring theory, with practical coloring algorithms under locality constraints.

Abstract

Let be a multigraph and be a list assignment on the edges of . Suppose additionally, for every vertex , the edges incident to have at least colors in common. We consider a variant of local edge-colorings wherein the color received by an edge must be contained in . The locality appears in the function , i.e., is some function of the local structure of in . Such a notion is a natural generalization of traditional local edge-coloring. Our main results include sufficient conditions on the function 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.
Paper Structure (4 sections, 6 theorems, 23 equations, 4 figures, 2 algorithms)

This paper contains 4 sections, 6 theorems, 23 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Proof of Theorem \ref{['theo:main_theo']}\ref{['theo:shannon']}
  4. Proof of Theorem \ref{['theo:main_theo']}\ref{['theo:vizing']}

Key Result

Theorem 1.4

Let $G$ be an $n$-vertex multigraph of maximum degree $\Delta$ and let $L \,:\,E(G) \to 2^{{\mathbb{N}}}$ be a list assignment for the edges of $G$. Furthermore, let $f(x, L)$ be as defined in eqn:local_list_size. Moreover, if $|f(x, L)| \leqslant \mathsf{poly}(\Delta, n)$ for each $x$, there is a $\mathsf{poly}(\Delta, n)$ time algorithm to compute such colorings.

Figures (4)

  • Figure 1: Shifting a coloring along a chain (Greek letters represent colors).
  • Figure 2: Path chains.
  • Figure 3: Shifting an initial segment of $P(e; \varphi, \alpha\beta)$ for $e \in E_G(x, y)$.
  • Figure 4: The process of shifting a fan.

Theorems & Definitions (16)

  • Definition 1.1
  • Definition 1.2: $L$-edge-coloring
  • Theorem 1.4
  • Proposition 1.5
  • proof
  • Corollary 1.6: to Theorem \ref{['theo:main_theo']}
  • Definition 2.1: Happy chains
  • Definition 2.2
  • Definition 2.4: ShannonChain
  • Lemma 2.5
  • ...and 6 more