Saturation Games for Odd Cycles

TL;DR

This paper investigates the $$-saturation game for odd cycles, establishing bounds on the maximum number of edges in the final graph when two players play optimally, extending understanding of saturation games for odd cycle families.

Contribution

It generalizes the $$-saturation game to families of odd cycles and provides bounds on the final graph's edge count for these families, advancing the theoretical understanding of saturation games.

Findings

Established lower bounds for $sat_g(; n)$ for families of odd cycles.

Provided upper bounds on $sat_g(; n)$ for specific odd cycle families.

Connected the bounds to the size of the odd cycle family and asymptotic behavior.

Abstract

Given a family of graphs $\mathcal{F}$, we consider the $\mathcal{F}$-saturation game. In this game 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 that lies 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 $sat_g(\mathcal{F};n)$ denote the number of edges that are in the final graph when both players play optimally. The $\{C_3\}$-saturation game was the first saturation game to be considered, but as of now the order of magnitude of $sat_g(\{C_3\},n)$ remains unknown. We consider a generalization of this game. Let $\mathcal{C}_{2k+1}:=\{C_3,\ C_5,\ldots,C_{2k+1}\}$. We prove that $sat_g(\mathcal{C}_{2k+1};n)\ge(\frac{1}{4}-\epsilon_k)n^2+o(n^2)$ for all $k\ge 2$ and that $sat_g(\mathcal{C}_{2k+1};n)\le (\frac{1}{4}-\epsilon'_k)n^2+o(n^2)$ for all $k\ge 4$, with $\epsilon_k<\frac{1}{4}$ and $\epsilon'_k>0$ constants tending to 0 as $k\to \infty$. In addition to this we prove $sat_g(\{C_{2k+1}\};n)\le \frac{4}{27}n^2+o(n^2)$ for all $k\ge 2$, and $sat_g(\mathcal{C}_\infty\setminus C_3;n)\le 2n-2$, where $\mathcal{C}_\infty$ denotes the set of all odd cycles.