On \(b\)-Coloring of Cartesian Product of Graphs

Abstract

A \(b\)-coloring of a graph \(G\) by \(k\) colors is a proper \(k\)-coloring of the vertices of \(G\) such that in each color class there exists a vertex having neighbors in all the other \(k-1\) color classes. The \(b\)-chromatic number \(\varphi(G)\) of a graph \(G\) is the maximum \(k\) for which \(G\) has a \(b\)-coloring by \(k\) colors. This concept was introduced by R.W. Irving and D.F. Manlove in \(1999\). In this paper, we study the \(b\)-chromatic numbers of the cartesian products of paths and cycles with complete graphs and the cartesian product of two complete graphs.