Game Chromatic Number of Cartesian Product Graphs

Abstract

The game chromatic number $\chi _{g}$ is considered for the Cartesian product $G\,\square \,H$ of two graphs $G$ and $H$. Exact values of $\chi _{g}(K_2\square H)$ are determined when $H$ is a path, a cycle, or a complete graph. By using a newly introduced "game of combinations" we show that the game chromatic number is not bounded in the class of Cartesian products of two complete bipartite graphs. This result implies that the game chromatic number $\chi_{g}(G\square H)$ is not bounded from above by a function of game chromatic numbers of graphs $G$ and $H$. An analogous result is derived for the game coloring number of the Cartesian product of graphs.

Game coloring the Cartesian product of graphs

“…If only the acyclic coloring method is used, the upper bound obtained is not as sharp as the one given in Theorem 1. The activation strategy alone simply does not work for the Cartesian product of graphs with large maximum degree, because an upper bound derived by using the activation strategy should be an upper bound for the game coloring number, but the game coloring number of K 1,n K 1,n is not bounded, as shown in [1]. Indeed, a result of Kierstead and Trotter [20] implies that col B g (K 1,n K 1,n ) is of the order of ln n. For a graph G, denote by sd 1 (G) the graph obtained from G by replacing each edge with a path of length 2.…”

“…It is shown in [1] that χ A g (K n,n K m,m ) goes to infinity if n, m goes to infinity. This shows that χ A g (G G ) is not bounded from above by any function of χ A g (G), χ A g (G ).…”

“…The game chromatic number of the Cartesian product of graphs was first studied in [1]. The exact values of χ A g (K 2 P n ) and χ A g (K 2 K n ) are determined.…”

“…The exact values of χ A g (K 2 P n ) and χ A g (K 2 K n ) are determined. It is also proved in [1] that the Cartesian product of two forests has game chromatic number at most 12 and the Cartesian product of two planar graphs has game chromatic number at most 650.…”

Game Chromatic Number of Generalized Petersen Graphs and Jahangir Graphs

“…In [4], Guan and Zhu introduced an upper bound to the game chromatic number of outerplanar graphs, which is χ g (OP) ≤ 7. In [5], Bartnicki et al studied the game chromatic number for the Cartesian product of two graphs. In [6], Sia studied the game for some families of Cartesian product graphs.…”