Local Nature of Brooks’ Colouring for Degree 3 Graphs

Abstract.

It is shown that for any two vertices u and v that are more than a constant distance apart in a 3-degree Brooks’ graph G, there exist two distinct 3-colourings of G of which one has u and v coloured the same, and the other has u and v coloured differently. Thus, in Brooks’ colouring, fixing the colour of one vertex does not fix the colour of vertices arbitrarily far away. In contrast, while 2-colouring a linked list, fixing the colour of one vertex fixes the colour of all others. This highly local nature of the problem can be seen as suggesting that, vis-a-vis the problem of 2-colouring a linked list, Brooks’ colouring may have a faster parallel algorithm.