The edge coloring of the Cartesian product of signed graphs

Abstract

According to Vizing's Theorem, a major question in the area of edge coloring is to determine whether a graph is Class 1 or 2. In 1984, Mohar proved that the Cartesian product $G\square H$ is Class 1 if G is Class 1 or both G and H have a perfect matching. Recently, Behr proved that the signed graph version of Vizing's Theorem: a signed graph $(G,\sigma )$ is either Class 1 or 2. Hence, we want to generalize Mohar's results to signed graphs. In this paper, we prove that $(G,\sigma )\square (H,\pi )$ is Class 1 if one of the factors, say $(G,\sigma )$, is Class 1 and there exists an edge coloring of $(G,\sigma )$ that satisfies a certain property, which is necessary as shown by an example. Let Δ-matching be a matching which covers every vertex of maximum degree. We also show that if both of $(G,\sigma )$ and $(H,\pi )$ have a Δ-matching and at least one of $\Delta \left(G\right),\Delta \left(H\right)$ is even, then $(G,\sigma )\square (H,\pi )$ is Class 1. This implies that if both of G and H have a Δ-matching, then $G\square H$ is Class 1, thereby slightly improving upon Mohar's results.