Chromatic Index of Signed Generalized Book Graphs and Signed Complete Graphs

A signed graph $(G,\sigma )$ consists of a graph $G$ and the signature $\sigma :E(G)\to \{+1,-1\}$. 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, $\gamma (v,e)\mathrm{\ne }\gamma (v,f)$. The chromatic index of a signed graph $(G,\sigma )$, denoted by ${\chi }^{\mathrm{\prime }}(G,\sigma )$, 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.