Game Chromatic Number of Honeycomb Related Networks

Document Type : Original paper


1 Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 15551, Al Ain, United Arab Emirates

2 Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan

3 Faculty of Education, University of the Ryukyus, Okinawa, Japan


Let $G$ be a simple connected graph having finite number of vertices (nodes). Let a coloring game is played on the nodes of $G$ by two players, Alice and Bob alternately assign colors to the nodes such that the adjacent nodes receive different colors with Alice taking first turn. Bob wins the game if he is succeeded to assign k distinct colors in the neighborhood of some vertex, where k is the available number of colors. Otherwise, Alice wins. The game chromatic number of G is the minimum number of colors that are needed for Alice to win this coloring game and is denoted by $\chi_{g}(G)$. In this paper, the game chromatic number $\chi_{g}(G)$ for some interconnecting networks such as infinite honeycomb network, elementary wall of infinite height and infinite octagonal network is determined. Also, the bounds for the game chromatic number $\chi_{g}(G)$ of infinite oxide network are explored.


Main Subjects

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