Towards an on-line version of Ohba’s conjecture

The on-line choice number of a graph is a variation of the choice number defined through a two person game. It is at least as large as the choice number for all graphs and is strictly larger for some graphs. In particular, there are graphs GG with |V(G)|=2χ(G)+1|V(G)|=2χ(G)+1 whose on-line choice numbers are larger than their chromatic numbers, in contrast to a recently confirmed conjecture of Ohba that every graph GG with |V(G)|⩽2χ(G)+1|V(G)|⩽2χ(G)+1 has its choice number equal to its chromatic number. Nevertheless, an on-line version of Ohba’s conjecture was proposed in [P. Huang, T. Wong and X. Zhu, Application of polynomial method to on-line colouring of graphs, European J. Combin., 2011]: every graph GG with |V(G)|⩽2χ(G)|V(G)|⩽2χ(G) has its on-line choice number equal to its chromatic number. This paper confirms the on-line version of Ohba’s conjecture for graphs GG with independence number at most 33. We also study list colouring of complete multipartite graphs K3⋆kK3⋆k with all parts of size 33. We prove that the on-line choice number of K3⋆kK3⋆k is at most 32k and present an alternate proof of Kierstead’s result that its choice number is ⌈(4k−1)/3⌉⌈(4k−1)/3⌉. For general graphs GG, we prove that if |V(G)|⩽χ(G)+χ(G) then its on-line choice number equals the chromatic number.