The Local Chromatic Number

Abstract

A graph vertex colouring is called k-local if the number of colours used in the closed neighbourhood of each vertex is at most k. The local chromatic number of a graph is the smallest k for which the graph has a proper k-local colouring. So unlike the chromatic number which is the minimum total number of colours required in a proper colouring, the local chromatic number is minimum number of colours that must appear in the closed neighbourhood of some vertex in a proper colouring.

In this thesis we will examine basic properties of the local chromatic number, and techniques used to determine or bound it. We will examine a theory that was sparked by Lovász's original proof of the Kneser conjecture, using topological tools to give lower bounds on the chromatic number, and see how it is applicable to give lower bounds on the local chromatic number as well.

The local chromatic number lies between the fractional chromatic number and the chromatic number, and thus it is particularly interesting to study when the gap between these two parameters is large. We will examine the local chromatic number for specific classes of graphs, and give a slight generalization of a result by Simonyi and Tardos that gives an upper bound on the local chromatic number for a class of graphs called Schrijver graphs.

Finally we will discuss open conjectures about the chromatic number and investigate versions adapted to the local chromatic number.