Summary: 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 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 the starting graph is connected and, after each player’s turn, the subgraph induced by the set of colored vertices is connected. The connected game chromatic number of a connected 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 connected outerplanar graph is at most 5 and that there exist connected outerplanar graphs with connected game chromatic number 4.
Moreover, we prove that for every integer \(k \geq 3\), there exist connected 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 connected bipartite graph.