On-Line List Colouring of Complete Multipartite Graphs

The Ohba Conjecture says that every graph G with |V (G)| ≤ 2χ(G) + 1 is chromatic choosable. This paper studies an on-line version of Ohba Conjecture. We prove that unlike the off-line case, for k ≥ 3, the complete multipartite graph K2?(k−1),3 is not on-line chromatic-choosable. Based on this result, the on-line version of Ohba Conjecture is modified as follows: Every graph G with |V (G)| ≤ 2χ(G) is on-line chromatic choosable. We present an explicit strategy to show that for any positive integer k, the graph K2?k is on-line chromatic-choosable. We then present a minimal function g for which the graph K2?(k−1),3 is on-line g-choosable.