Total Colourings - A survey

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 $\mathrm{\Delta }(G)+2$ colors, where $\mathrm{\Delta }(G)$ 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.