(2, t)-choosable graphs

A (k, t)-list assignment L of a graph G assigns a list of k colors available at each vertex v in G and vertical bar boolean OR(v is an element of V(G)) L(v)vertical bar = t. An L-coloring is a proper coloring c such that c(v) is an element of L(v) for each v is an element of V (G). A graph G is (k, t)-choosable if G has an L-coloring for every (k, t)-list assignment L. Erdos, Rubin, and Taylor proved that a graph is (2, t)-choosable for any t >= 2 if and only if a graph does not contain some certain subgraphs. Chare-onpanitseri, Punnim, and Uiyyasathian proved that an n-vertex graph is (2, t)-choosable for 2n - 6 <= t <= 2n - 4 if and only if it is triangle-free. Furthermore, they proved that a triangle-free graph with n vertices is (2,2n - 7)-choosable if and only if it does not contain K-3,K-3 - e where e is an edge. Nakprasit and Ruksasakchai proved that an n-vertex graph G that does not contain C-5 V Kk-2 and Kk+1 for k >= 3 is (k,kn - k(2) - 2k)-choosable. For a non-2-choosable graph G, we find the minimum t(1) >= 2 and the maximum t(2) such that the graph G is not (2, t(i))-choosable for i = 1,2 in terms of certain subgraphs. The results can be applied to characterize (2, t)-choosable graphs for any t.