Total Colourings - A survey

TL;DR

This survey reviews the concept of total coloring in graphs, discussing the total chromatic number, key conjectures like Vizing's, and the current state of research, especially for planar graphs.

Contribution

It provides a comprehensive overview of total coloring theory, conjectures, and open problems, highlighting recent developments and unresolved questions.

Findings

Vizing's conjecture remains unproven for many graph classes.

The total chromatic number is closely related to the maximum degree of the graph.

Open problems include the total coloring of planar graphs.

Abstract

The smallest integer $k$ needed for the assignment of colors to the elements so that the coloring is proper (vertices and edges) is called the total chromatic number of a graph. Vizing and Behzed conjectured that the total coloring can be done using at most $\Delta(G)+2$ colors, where $\Delta(G)$ is the maximum degree of $G$. It is not settled even for planar graphs. In this paper we give a survey on total coloring of graphs.