The b-fold indicated L-coloring game on G is played by two players: Ann and Ben, where G is a graph and L is a list assignment of G. In each round, Ann chooses an uncolored vertex v, and Ben colors v with a b-set ϕ (v) from L(v) such that none of the colors in ϕ (v) have been used by its colored neighbors. If all vertices are colored, Ann wins the game. Otherwise, after some rounds, there is an uncolored vertex v with less than b available colors (i.e., colors in its list not used by its colored neighbors), and Ben wins the game. We say G is indicated (L, b)-colorable if Ann has a winning strategy for the b-fold indicated L-coloring game. For a mapping g: V(G) →ℕ , we say G is indicated (g, b)-choosable if G is indicated (L, b)-colorable for every list assignment L of G with |L(v)|≥ g(v) for each vertex v. If g(v)=a for every vertex v, then indicated (g, b)-choosable is called indicated (a, b)-choosable. The indicated choice number ch_i(G) is the least integer k such that G is indicated (k, 1)-choosable (also called indicated k-choosable). The fractional indicated choice number of G is ch_i^f(G)=inf{a/b:G is indicated (a,b)-choosable} . This paper proves that for any finite graph G, ch_i^f(G)=ch_i(G) ; a connected graph G is indicated 2-choosable if and only if its core is K_1 or Θ _2,2,2 or an even cycle; for m ≥ 2 , a graph G is indicated (2m, m)-choosable if and only if G is a tree. A graph G is called indicated k-choosable-critical if G is not indicated (k-1) -choosable, but any proper subgraph is indicated (k-1) -choosable. We give a characterization of indicated 3-choosable critical graphs.
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