Linear Bounds for Cyclefree Saturation Games
Abstract
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 nontrivial 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.05295
 Bibcode:
 2021arXiv210805295E
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics;
 05C57
 EPrint:
 18 pages, 2 figures