On Caterpillars of 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. If at any point there is an uncolored vertex without available color, then Bob wins. The game chromatic number of G is the smallest number t such that Alice has a winning strategy. It is known that for forests this number is at most 4. We find exact values for the game chromatic number of an infinite subclass of forests (composed by caterpillars), in order to contribute to the open problem of characterizing forests with different game chromatic numbers. Moreover, we show two sufficient conditions and two necessary conditions for any tree and caterpillar to have game chromatic number 4, respectively.