On-Line List Colouring of Random Graphs

Abstract

The on-line list colouring of binomial random graphs $\mathcal{G}(n,p)$ is studied. We show that the on-line choice number of $\mathcal{G}(n,p)$ is asymptotically almost surely asymptotic to the chromatic number of $\mathcal{G}(n,p)$, provided that the average degree d = p(n − 1) tends to infinity faster than (loglogn)^1∕3(logn)² n ^2∕3. For sparser graphs, we are slightly less successful; we show that if d ≥ (logn)^2+ɛ for some ɛ > 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.