The Game Coloring Number of Planar Graphs

Abstract

This paper discusses a variation of the game chromatic number of a graph: the game coloring number. This parameter provides an upper bound for the game chromatic number of a graph. We show that the game coloring number of a planar graph is at most 19. This implies that the game chromatic number of a planar graph is at most 19, which improves the previous known upper bound for the game chromatic number of planar graphs. Academic Press

Game chromatic number of outerplanar graphs

“…Recently, X. Zhu has proved that the game chromatic number of a planar graph is at most 19 [5], the game chromatic number of a partial k-tree is at most 3k + 2 [6], and that, for g ≥ 1, any graph embeddable on the orientable surface of genus g has game chromatic number at most [7].…”

The game Grundy number of graphs

“…The game colouring number of a graph was first formally introduced in [17] as a tool in the study of the game chromatic number of graphs. However, it is of independent interest.…”

Structural Properties of Sparse Graphs

“…It was proved by Faigle et al [47] that the game coloring number of a forest is at most 4, and that the game coloring number of an interval graph G is at most 3ω−2. It was proved by Zhu [118] that the game coloring number of the planar graphs is at most 19 and this bound has been further reduced by Kierstead to 18 [62] and by Zhu to 17 [121]. It has also been shown by Guan and Zhu [55] that the outerplanar graphs have game coloring number at most 7.…”