Adjacent Vertex Distinguishing Total Colorings of Graphs with Four Distinguishing Constraints

Abstract

The frequency assignment problem originated in researching mobile communication networks. A proper total coloring of a graph \(G\) is a coloring of both edges and vertices of \(G\) such that no two adjacent or incident elements receive the same color. As known, the vertex distinguishing total coloring is one of the suitable tools for investigating the frequency assignment problem. We introduce a new graph total coloring, called \((4)\)-adjacent vertex distinguishing total coloring (\((4)\)-AVDTC), in this paper. Our coloring contains the adjacent vertex distinguishing total coloring. The minimum number of colors required for every \((4)\)-AVDTC of \(G\) is called the \((4)\)-AVDTC chromatic number of \(G\). We will show that using at most \(\Delta(G) + 4\) colors can achieve at least \(4\) different adjacent vertex distinguishing actions for some communication networks \(G\). The exact \((4)\)-AVDTC chromatic numbers of several classes of graphs are determined here and a problem is presented.