The coloring game on planar graphs with large girth, by a result on sparse cactuses

One can ask if this bound of 5 can be improved for a sufficiently large girth. In this paper, we prove that it cannot. More than that, we prove that there are cactuses CT (i.e. graphs in which each edge only belongs to at most one cycle) having χg(CT)=5 despite having arbitrary large girth, and even arbitrary large distance between its cycles.