Edge Coloring Signed Graphs

TL;DR

This paper introduces a new method for edge coloring signed graphs that generalizes classical edge coloring, maintains desirable properties, and extends Vizing's Theorem to signed graphs.

Contribution

It defines a proper edge coloring for signed graphs, relates it to vertex coloring of line graphs, and proves an upper bound on the number of colors needed, extending Vizing's Theorem.

Findings

Proper edge coloring for signed graphs is defined.

The method generalizes classical edge coloring.

An upper bound of Δ+1 colors is established, extending Vizing's Theorem.

Abstract

We define a method for edge coloring signed graphs and what it means for such a coloring to be proper. Our method has many desirable properties: it specializes to the usual notion of edge coloring when the signed graph is all-negative, it has a natural definition in terms of vertex coloring of a line graph, and the minimum number of colors required for a proper coloring of a signed simple graph is bounded above by {\Delta} + 1 in parallel with Vizing's Theorem. In fact, Vizing's Theorem is a special case of the more difficult theorem concerning signed graphs.