Towards on-line Ohba's conjecture

Ohba conjectured that every graph G with |V (G)|6 2χ(G)+1 has its choice number equal its chromatic number. The on-line choice number of a graph is a variation of the choice number defined through a two person game, and is always at least as large as its choice number. Based on the result that for k > 3, the complete multipartite graph K2?(k−1),3 is not on-line k-choosable, the on-line version of Ohba’s conjecture is modified in [P. Huang, T. Wong and X. Zhu, Application of polynomial method to on-line colouring of graphs, European J. Combin., 2011] as follows: Every graph G with |V (G)| 6 2χ(G) has its on-line choice number equal its chromatic number. In this paper, we prove that for any graph G, there is an integer n such that the join G + Kn of G and Kn has its on-line choice number equal chromatic number. Then we show that the on-line version of Ohba conjecture is true if G has independence number at most 3. We also present an alternative proof of the result that Ohba’s conjecture is true for graphs of independence number at most 3 and an alternative proof of the following result of Kierstead: For any positive integer k, the complete multipartite graph K3?k has choice number d(4k−1)/3e. Finally, we prove that the on-line choice number of K3?k is at most 3 2 k. The exact value of the on-line choice number of K3?k remains unknown.