  The graph coloring game is a two-player game in which, given a graph G and a
set of k colors, the two players, Alice and Bob, take turns coloring properly
an uncolored vertex of G, Alice having the first move. Alice wins the game if
and only if all the vertices of G are eventually colored. The game chromatic
number of a graph G is then defined as the smallest integer k for which Alice
has a winning strategy when playing the graph coloring game on G with k colors.
In this paper, we introduce and study a new version of the graph coloring game
by requiring that, after each player's turn, the subgraph induced by the set of
colored vertices is connected. The connected game chromatic number of a graph G
is then the smallest integer k for which Alice has a winning strategy when
playing the connected graph coloring game on G with k colors. We prove that the
connected game chromatic number of every outerplanar graph is at most 5 and
that there exist outerplanar graphs with connected game chromatic number 4.
Moreover, we prove that for every integer k $\ge$ 3, there exist bipartite
graphs on which Bob wins the connected coloring game with k colors, while Alice
wins the connected coloring game with two colors on every bipartite graph.
