Azuma’s Inequality and a Strengthening of Brooks’ Theorem

Abstract

In this chapter, we introduce a new tool for proving bounds on concentration. It differs from the tools we have mentioned so far, in that it can be applied to a sequence of dependent trials. To see a concrete example of such a situation, imagine that we are colouring the vertices of a graph one by one, assigning to each vertex a colour chosen uniformly from those not yet assigned to any of its coloured neighbours. This ensures that the colouring obtained is indeed a proper colouring, and analyzing such a random process may yield good bounds on the minimum number of colours required to obtain vertex colourings with certain properties. However, our choices at each vertex are now no longer independent of those made at the other vertices.