The Smarandachely Adjacent Vertex Total Coloring of a kind of \(3\)-regular Graph

Abstract

A proper total coloring of a graph \(G\) is called Smarandachely adjacent vertex total coloring of graph if for any two adjacent and distinct vertices \(u\) and \(v\) in \(G\), the set of colors assigned to the vertices and the edges incident to \(u\) doesn’t contain the set of colors assigned to the vertices and the edges incident to \(v\), vice versa. The minimal number of colors required for a Smarandachely adjacent vertex total coloring of graph is called the Smarandachely adjacent vertex total chromatic number of graph. In this paper, we define a kind of \(3\)-regular Multilayer Cycle \(Re(n,m)\) and obtain the Smarandachely adjacent vertex total chromatic number of it.