Chromatic Index of Signed Generalized Book Graphs and Signed Complete Graphs

Abstract

A signed graph $(G,\sigma)$ consists of a graph $G$ and the signature $\sigma : E(G) \rightarrow \{+1,-1\}$. An incidence of $G$ is a pair $(v,e)$, where $v$ is one of the end vertices of an edge $e \in E(G)$. A proper $q$-edge coloring $\gamma$ of signed graph $(G,\sigma)$ is an assignment of colors to incidences satisfying that $\gamma(v,e) = - \sigma(e) \gamma(w,e)$ for every edge $e=vw$ and for any two incidences $(v,e)$ and $(v,f)$, involving the same vertex, $\gamma(v,e) \neq \gamma(v,f)$. The chromatic index of a signed graph $(G,\sigma)$, denoted by $\chi'(G,\sigma)$, is the minimum number $q$ for which $(G,\sigma)$ has a proper $q$-edge coloring. In this paper, we determine the chromatic index of signed generalized book graphs. We also determine the chromatic index of signed complete graphs of order up to six.