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

Abstract

A \(b\)-coloring of a graph \(G\) with \(k\) colors is a proper coloring of \(G\) using \(k\) colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer \(k\) for which \(G\) has a \(b\)-coloring using \(k\) colors is the \(b\)-chromatic number \(\beta(G)\) of \(G\). The \(b\)-spectrum \(\mathcal{S}_b(G)\) of a graph \(G\) is the set of positive integers \(k\), \(\chi(G) \leq k \leq b(G)\), for which \(G\) has a \(b\)-coloring using \(k\) colors. A graph \(G\) is \(b\)-continuous if \(\mathcal{S}_b(G) = \{\chi(G), \ldots, b(G)\}\). It is known that for any two graphs \(G\) and \(H\), \(b(G \Box H) \geq \max\{b(G), b(H)\}\), where \(\Box\) stands for the Cartesian product. In this paper, we determine some families of graphs \(G\) and \(H\) for which \(b(G \Box H) \geq b(G) + b(H) – 1\). Further, we show that if \(O_k,i=1,2,\ldots,n\) are odd graphs with \(k_i \geq 4\) for each \(i\), then \(O_{k_1} \Box O_{k_2} \Box \ldots \Box O_{k_n}\) is \(b\)-continuous and \(b(O_{k_1} \Box O_{k_2} \Box \ldots \Box O_{k_n}) = 1 + \sum\limits_{i=1}^{n} k_i\).