On-line list colouring of random graphs

TL;DR

This paper investigates the on-line list colouring of random graphs G(n,p), showing that the on-line choice number closely matches the chromatic number in dense graphs and providing bounds for sparser graphs.

Contribution

It establishes asymptotic equivalence of on-line choice number and chromatic number for dense random graphs and provides bounds for sparser cases.

Findings

On-line choice number asymptotically equals chromatic number for dense graphs.

For graphs with average degree d > (log n)^(2+epsilon), the on-line choice number is within a constant factor of the chromatic number.

For graphs with constant average degree, the on-line choice number is at most a constant factor larger than the chromatic number.

Abstract

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 log n)^1/3(log n)^2n^(2/3). For sparser graphs, we are slightly less successful; we show that if d>(log n)^(2+epsilon) for some epsilon>0, then the on-line choice number is larger than the chromatic number by at most a multiplicative factor of C, where C in [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.