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 \(\phi (v)\) from L(v) such that none of the colors in \(\phi (v)\) 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 \(g: V(G) \rightarrow \mathbb {N}\) , we say G is indicated (g, b)-choosable if G is indicated (L, b)-colorable for every list assignment L of G with \(|L(v)|\ge g(v)\) 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 \(ch_i^f(G)=\inf \{\frac{a}{b}:G\text { is indicated } (a,b)\text {-choosable}\}\) . 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 \( \Theta _{2,2,2}\) or an even cycle; for \( m \ge 2\) , 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.
