Paintability of complete bipartite graphs.

Paintability of graphs, which is also called online choosability of graphs, was introduced by Schauz and by Zhu in 2009, and has been studied in point of the difference from choosability of graphs. Compared with the choice number and the chromatic number, it is much more difficult to determine the paint number of a given graph. For example, even the 3-paintable complete bipartite graphs have not been characterized yet. In this paper, we focus on the paintability of complete bipartite graphs, especially m -paintability of K m + 1 , r. Let r OL (m) denote the least integer r such that K m + 1 , r is not m -paintable. In this paper, we determine the order of r OL (m) as r OL (m) = Θ (m m − 1). Let r (m) denote the least r such that K m + 1 , r is not m -choosable, for which Hoffman and Johnson Jr. determined as r (m) = m m − (m − 1) m = Θ (m m). As a corollary, we can show that r OL (m) / r (m) tends to 0 when m goes to infinity. [ABSTRACT FROM AUTHOR]