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 (loglogn)1/3(logn)2n2/3. For sparser graphs, we are slightly less successful; we show that if d≥(logn)2+ϵ 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.