This theorem is a stronger result about an upper bound for the chromatic number due to Rowland Leonard Brooks (1916 – 1993).

Theorem: Brooks' Theorem

Let $G$ be a connected simple graph with the maximum vertex degree $d.$ If $G$ is neither a cycle with an odd number of vertices, nor a complete graph, then the chromatic number of $G$ obeys the following inequality: $$\chi(G)\le d.$$

Proofs: 1

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  1. Aldous Joan M., Wilson Robin J.: "Graphs and Applications - An Introductory Approach", Springer, 2000