General Vertex-Distinguishing Total Coloring of Complete Bipartite Graphs

Abstract

The general vertex-distinguishing total chromatic number of a graph \(G\) is the minimum integer \(k\), for which the vertices and edges of \(G\) are colored using \(k\) colors such that there are no two vertices possessing the same color-set, where a color-set of a vertex is a set of colors of the vertex and its incident edges. In this paper, we discuss the general vertex-distinguishing total chromatic number of complete bipartite graphs \(K_{m,n}\), and obtain the exact value of this number for some cases in terms of \(m\) and \(n\). Particularly, we give the bounds of this number for \(K_{n,n}\).