Total coloring

Summary

In graph theory, total coloring is a type of graph coloring on the vertices and edges of a graph. When used without any qualification, a total coloring is always assumed to be proper in the sense that no adjacent edges, no adjacent vertices and no edge and either endvertex are assigned the same color. The total chromatic number $\chi''(G)$ of a graph $G$ is the fewest colors needed in any total coloring of $G$.