Edge coloring signed graphs

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 allnegative, 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 ∆ + 1 in parallel with Vizing's Theorem. In fact, Vizing's Theorem is a special case of the more difficult theorem concerning signed graphs.

Edge coloring of graphs of signed class 1 and 2

“…This implies that they are 2-regular, spanning and balanced -since otherwise they could not be colored using only colors {−k, k}. This also implies that our matching has 1 2 n(G) edges, which means it is perfect.…”

“…Behr [4] defined class ratio C(G) of graph G to be the number of possible signatures σ : E(G) → {±1} such that the signed graph (G, σ) is ∆(G)-colorable, divided by the number of all possible signatures defined on E(G), i.e. 2 m(G) .…”

“…, v 1 1 , v, v 1 2 span a path in S 0 and its edges can be colored with colors ±1. We can assume that incidence u : uu 1 3 gets color 1. Let us observe that the remaining edges also span a path, end its edges can be colored with colors ±2.…”

“…In 2020, Behr [1] introduced a concept of an edge coloring of signed graphs as a generalization of ordinary graph edge coloring. Let n be a positive integer and…”

“…Behr [1] proved that every signed path can be colored using 2 colors and a signed cycle can be colored with 2 colors if and only if it's balanced. The main result of Behr's article [1] is the generalized Vizing's theorem. We will call it the Behr's theorem to emphasize the importance of Behr's achievement.…”