Chromatic Index of Signed Generalized Book Graphs and Signed Complete Graphs

TL;DR

This paper investigates the chromatic index of signed generalized book graphs and signed complete graphs up to order six, providing exact values and extending understanding of edge coloring in signed graphs.

Contribution

It determines the chromatic index for specific classes of signed graphs, namely generalized book graphs and complete graphs up to six vertices, which was previously unknown.

Findings

Chromatic index of signed generalized book graphs is established.

Chromatic index of signed complete graphs up to order six is determined.

Provides formulas and methods for calculating edge colorings in signed 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.