On Edge Coloring of Signed Generalized Book Graphs and Signed Complete Graphs

Abstract

A signed graph [Formula: see text] consists of graph [Formula: see text] and signature [Formula: see text]. An incidence of [Formula: see text] is a pair [Formula: see text], where [Formula: see text] is one of the end vertices of edge [Formula: see text]. A proper [Formula: see text]-edge coloring [Formula: see text] of the signed graph [Formula: see text] is an assignment of colors to incidences satisfying that [Formula: see text] for every edge [Formula: see text] and for any two incidences [Formula: see text] and [Formula: see text], involving the same vertex, [Formula: see text]. The chromatic index of a signed graph [Formula: see text], denoted by [Formula: see text], is the minimum number [Formula: see text] for which [Formula: see text] has a proper [Formula: see text]-edge coloring. In this paper, we consider the edge coloring of signed generalized book graphs and signed complete graphs. We first classified the switching non-isomorphic signed generalized book graphs and then computed the chromatic index of all signed generalized book graphs. We also determined the chromatic index of the signed complete graphs of up to six order.