---
tag: HR54EB7CCY
kind: concept
name: Total coloring
aliases: [total-chromatic-number, total-coloring-conjecture, total-graph]
sources:
  - wikipedia: en:690702@r1285190354
    locator: Total coloring
    note: adopted
    mention: wikipedia://en/690702
    sections: [Summary]
same_as:
  wikidata: Q7828102
  wikipedia: Total coloring
inventory: [KZA01HA8S4]
---

## Summary

In graph theory, **total coloring** is a type of [[graph coloring|0NCPXH194Y]] 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 $G$.