chromatic number and girth

A famous theorem of P. Erdős11See the very readable P. Erdős, and probability, Canad J. Math. 11 (1959), 34–38..

Theorem 1

For any natural numbersMathworldPlanetmath k and g, there exists a graph G with chromatic numberMathworldPlanetmath χ(G)k and girth girth(G)g.

Obviously, we can easily have graphs with high chromatic numbers. For instance, the complete graphMathworldPlanetmath Kn trivially has χ(Kn)=n; however girth(Kn)=3 (for n3). And the cycle graph Cn has girth(Cn)=n, but

χ(Cn)={1n=12n even3otherwise.

It seems intuitively plausible that a high chromatic number occurs because of short, “local” cycles in the graph; it is hard to envisage how a graph with no short cycles can still have high chromatic number.

Instead of envisaging, Erdős’ proof shows that, in some appropriately chosen probability spaceMathworldPlanetmath on graphs with n vertices, the probability of choosing a graph which does not have χ(G)k and girth(G)g tends to zero as n grows. In particular, the desired graphs exist.

This seminal paper is probably the most famous application of the probabilistic method, and is regarded by some as the foundation of the method.22However, as always, with the benefit of hindsight we can see that the probabilistic method had been used before, e.g. in various applications of Sard’s theorem. This does nothing to diminish from the importance of the clear statement of the tool. Today the probabilistic method is a standard tool for combinatorics. More constructive methods are often preferred, but are almost always much harder.

Title chromatic number and girth
Canonical name ChromaticNumberAndGirth
Date of creation 2013-03-22 12:46:03
Last modified on 2013-03-22 12:46:03
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 7
Author mathcam (2727)
Entry type Theorem
Classification msc 05C15
Classification msc 05C38
Classification msc 05C80
Related topic Girth
Related topic ChromaticNumber