Sharp concentration of the equitable chromatic number of dense random graphs

Anequitable colouringof a graph G is a vertex colouring where no two adjacent vertices are coloured the same and, additionally, the colour class sizes differ by at most 1. Theequitable chromatic number chi=(G) is the minimum number of colours required for this. We study the equitable chromatic number of the dense random graph G(n, m) where m = left perpendicualarp((n)(2))right perpendicualar and 0 < p < 0.86 is constant. It is a well-known question of Bollobas [3] whether for p = 1/2 there is a function f (n) -> infinity such that, for any sequence of intervals of length f (n), the normal chromatic number of G(n, m) lies outside the intervals with probability at least 1/2 if n is large enough. Bollobas proposes that this is likely to hold for f (n) = log n. We show that for the equitable chromatic number, the answer to the analogous question is negative. In fact, there is a sub-sequence (nj)(j is an element of N) of the integers where chi=(G(n(j), m(j))) is concentrated on exactly one explicitly known value. This constitutes surprisingly narrow concentration since in this range the equitable chromatic number, like the normal chromatic number, is rather large in absolute value, namely asymptotically equal to n/(2 log(b) n) where b = 1/(1- p).