On-line List Colouring of Random Graphs

In this paper, the on-line list colouring of binomial random graphs G (n, p) is studied. We show that the on-line choice number of G (n, p) is asymptotically almost surely asymptotic to the chromatic number of G (n, p), provided that the average degree d = p(n − 1) tends to infinity faster than (log logn)(logn)n. For sparser graphs, we are slightly less successful; we show that if d ≥ (logn) for some ε > 0, then the on-line choice number is larger than the chromatic number by at most a multiplicative factor of C, where C ∈ [2, 4], depending on the range of d. Also, for d = O(1), the on-line choice number is by at most a multiplicative constant factor larger than the chromatic number.