Supermodular Extension of Vizing's Edge-Coloring Theorem

TL;DR

This paper introduces a generalized framework extending Vizing's edge-coloring theorem using intersecting 2/3-supermodular functions, unifying and broadening classical results in graph coloring.

Contribution

It presents a new generalization of Vizing's theorem through intersecting 2/3-supermodular functions and offers an alternative proof of Gupta's edge-coloring theorem.

Findings

Unified framework for edge-coloring theorems

Introduction of intersecting 2/3-supermodular functions

Alternative proof of Gupta's theorem

Abstract

K\H{o}nig's edge-coloring theorem for bipartite graphs and Vizing's edge-coloring theorem for general graphs are celebrated results in graph theory and combinatorial optimization. Schrijver generalized K\H{o}nig's theorem to a framework defined with a pair of intersecting supermodular functions. The result is called the supermodular coloring theorem. This paper presents a common generalization of Vizing's theorem and a weaker version of the supermodular coloring theorem. To describe this theorem, we introduce intersecting 2/3-supermodular functions, which are extensions of intersecting supermodular functions. The paper also provides an alternative proof of Gupta's edge-coloring theorem using a special case of this supermodular version of Vizing's theorem.