Adjacent-vertex-distinguishg-total-coloring of graphs

Abstract

This is a research project to request a master scholarship for Atílio Gomes Luiz. It is proposed to investigate a total-coloring variant called adjacent-vertex-distinguishing-total-coloring(AVD-total-coloring). Given a total-coloring, we label each vertex of a graph with the set of colors appearing on the vertex and incidentedges. When these sets are distinct for any two adjacent vertices, the total coloring is an AVD-total-coloring. The smallest number of colors for which a graph admits an AVD-total-coloring is the AVD-total-chromatic number. In the seminal article, the authors determined the AVD-total-chromatic number for few classes of graphs. Additionaly, they have conjectured a bound in terms of the maximum degree of a graph. The goal of this project is to investigate this conjecture for given positive evidence for its validity and, if possible, to determine the AVD-total-chromatic number for the graphs of some classes.