On Edge Cover Coloring of Join Graphs

Abstract

A join graph is the complete union of two arbitrary graphs. An edge cover coloring is a coloring of edges of \(E(G)\) such that each color appears at each vertex \(v \in V(G)\) at least one time. The maximum number of colors needed to edge cover color \(G\) is called the edge cover chromatic index of \(G\) and denoted by \(\chi’C(G)\). It is well known that any simple graph \(G\) has the edge cover chromatic index equal to \(\delta(G)\) or \(\delta(G) – 1\), where \(\delta(G)\) is the minimum degree of \(G\). If \(\chi’C(G) = \delta(G)\), then \(G\) is of C1-Class , otherwise \(G\) is of C2-Class . In this paper, we give some sufficient conditions for a join graph to be of C1-Class.