New $\mathcal{F}$-Saturation Games on Directed Graphs

Lee, Jonathan D.; Riet, Ago-Erik

doi:10.48550/arxiv.1409.0565

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2022

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“…Saturation games have also been generalized to other combinatorial structures such as hypergraphs and directed graphs [10,12].…”

Section: Introductionmentioning

confidence: 99%

Linear Bounds for Cycle-Free Saturation Games

English

Masařík

McCourt

et al. 2022

Electron. J. Combin.

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Given a family of graphs $\mathcal{F}$, we define the $\mathcal{F}$-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 $\mathcal{F}$. 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 $\textrm{sat}_g(n,\mathcal{F})$ 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 $\textrm{sat}_g(n,\mathcal{F})$. In this work, we find collections of infinite families of cycles $\mathcal{C}$ such that $\textrm{sat}_g(n,\mathcal{C})$ has linear growth rate.

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“…Saturation games have also been generalized to other combinatorial structures such as hypergraphs and directed graphs [10,12].…”

Section: Introductionmentioning

confidence: 99%