Linear Bounds for Cycle-free Saturation Games

08/11/2021

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by   Sean English, et al.

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Iowa State University of Science and Technology

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University of Illinois at Urbana-Champaign

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Charles University in Prague

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Given a family of graphs ℱ, we define the ℱ-saturation game as follows. Two players alternate adding edges to an initially empty graph on n vertices, with the only constraint being that neither player can add an edge that creates a subgraph in ℱ. The game ends when no more edges can be added to the graph. One of the players wishes to end the game as quickly as possible, while the other wishes to prolong the game. We let sat_g(n,ℱ) denote the number of edges that are in the final graph when both players play optimally. In general there are very few non-trivial bounds on the order of magnitude of sat_g(n,ℱ). In this work, we find collections of infinite families of cycles 𝒞 such that sat_g(n,𝒞) has linear growth rate.