On Adjacent Vertex-Distinguishing Total Chromatic Number of Generalized Petersen Graphs

Abstract

Analyzing chromatic number in coloring problem is a tough topic in graph analysis. We focus on the basic theory for a particular type of chromatic number. This will give us insights on the basic topological structure guiding lots of networks in the coming trend of big data era. An adjacent vertex-distinguishing total k-coloring is a proper total k-coloring of a graph G such that for any two adjacent vertices, the set of colors appearing on the vertex and its incident edges are different. The smallest k for which such a coloring of G exists is called the adjacent vertex-distinguishing total chromatic number, and denoted by ?at(G). It has been proved that if the graph G satisfies ?(G)=3, then ?at(G)= 6. However, it is very difficult to determine whether ?at(G)= 5. In this paper, we focus on a special class of 3-regular graphs, the generalized Petersen graphs P(n, k), and show that ?at(P(n, k)) = 5, which improves the bound ?at (P(n, k))= 6.