Brooks' theorem

Builds on Maximum degree Chromatic number

Statement

Let $G$ be a finite connected simple graph with maximum degree $\Delta$ . Suppose that $G$ is not a complete graph and not a cycle graph of odd length. Then $\chi(G) \le \Delta$ .

Remarks

A greedy colouring gives $\chi(G) \le \Delta + 1$ for every graph. Brooks' theorem improves this bound by $1$ in all other cases. The two exceptions need $\Delta + 1$ colours: $\chi(K_n) = n$ with $\Delta = n - 1$ , and an odd cycle has $\chi = 3$ with $\Delta = 2$ .

History

R. Leonard Brooks published the theorem in 1941.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Graph definition
Graph colouring definition
Neighbourhood definition
K-colourable graph definition
Degree of a vertex definition
Chromatic number definition
Maximum degree definition
Brooks' theorem statement

Tag	`0Y4TV2D5RQ`
URI	`m2://0Y4TV2D5RQ`
Kind	statement, theorem
Writing	W1 stated
MSC	05C15
Status	This copy of the site has no build data. Run `m2 status 0Y4TV2D5RQ`.
Uses	Maximum degree, Chromatic number
Same as	Wikidata Q512897 Wikipedia Brooks' theorem

Brooks' theorem

Statement

Remarks

History

Reading path

Further reading

What this page needs for W2

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