Some results on (a:b)(a:b)-choosability

A solution to a problem of Erdős, Rubin and Taylor is obtained by showing that if a graph GG is (a:b)(a:b)-choosable, and c/d>a/bc/d>a/b, then GG is not necessarily (c:d)(c:d)-choosable. Applying probabilistic methods, an upper bound for the kkth choice number of a graph is given. We also prove that a directed graph with maximum outdegree dd and no odd directed cycle is (k(d+1):k)(k(d+1):k)-choosable for every k≥1k≥1. Other results presented in this article are related to the strong choice number of graphs (a generalization of the strong chromatic number). We conclude with complexity analysis of some decision problems related to graph choosability.