First

this paper introduces a new two-person game coloring

game chromatic and game chromatic number.Their difference is compared

and the properties of the game chromatic number Ⅱ are discussed.Second

on the basis of the path and cycle

the game chromatic numbers of θ-graph and general θ-graph are obtained respectively

and the winning strategy for A player is presented.Finally

the results are extended to obtain the game chromatic number of even θ-graph

and the winning strategy for A player is given.