On the degree of trees with game chromatic number 4

The coloring game is played by Alice and Bob on a finite graph G. They take turns properly coloring the vertices with t colors. The goal of Alice is to color the input graph with t colors, and Bob does his best to prevent it. If at any point there exists an uncolored vertex without available color, then Bob wins; otherwise Alice wins. The game chromatic number chi g(G) of G is the smallest number t such that Alice has a winning strategy. In 1991, Bodlaender showed the smallest tree T with chi g(T) equal to 4, and in 1993 Faigle et al. proved that every tree T satisfies the upper bound chi g(T)<= 4. The stars T = K1,p with p >= 1 are the only trees satisfying chi g(T) = 2; and the paths T = Pn, n >= 4, satisfy chi g(T) = 3. Despite the vast literature in this area, there does not exist a characterization of trees with chi g(T) = 3 or 4. We answer a question about the required degree to ensure chi g(T) = 4, by exhibiting infinitely many trees with maximum degree 3 and game chromatic number 4.