Linear Bounds for Cycle-free Saturation Games

TL;DR

This paper investigates the cycle-free saturation game, establishing linear bounds on the number of edges in the final graph for certain cycle collections, advancing understanding of game dynamics in extremal graph theory.

Contribution

The work identifies infinite families of cycles for which the saturation game has linear growth bounds, providing new bounds in the study of saturation games.

Findings

Established linear growth bounds for specific cycle collections

Provided new insights into the extremal properties of saturation games

Extended the understanding of game duration in cycle-avoidance scenarios

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 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.