The game coloring number of pseudo partial k-trees

Abstract

This paper introduces a new class of graphs: ( a, b)-pseudo partial k-trees. In some sense, the parameters ( a, b) measure a graph's deviation from being a partial k-tree. In particular, a (0,0)-pseudo partial k-tree is just a partial k-tree. We discuss the game coloring number (as well as the game chromatic number) of ( a, b)-pseudo partial k-trees, and prove that the game coloring number of an ( a, b)-pseudo partial k-tree is at most 3 k+2 a+ b+2. In particular, the game coloring number of a partial k-tree is at most 3 k+2. This reduces considerably the previous known upper bound for the game chromatic number of a partial k-tree. Then we describe another strategy for Alice for playing the game, which gives a better upper bound for the game coloring number of ( a, b)-pseudo partial 2-trees. Namely, we prove that the game coloring number of an ( a, b)-pseudo partial 2-tree is at most a+ b+8. By using this result, we prove that the game coloring number of a graph embeddable on an orientable surface of genus g⩾1 is at most ⌊ 1 2 (3 1+48g +23)⌋. This is the first upper bound for the game coloring number of such graphs, and it also improves considerably the previous known upper bound for the game chromatic number of such graphs. Moreover, the strategy for Alice introduced in this paper for playing the game can be considered as a common generalization of the previous known strategies for playing the game on forests, interval graphs, outerplanar graphs and planar graphs.