The Game Saturation Number of a Graph

TL;DR

This paper investigates the game saturation number in various graph settings, determining exact values and bounds for different graph families and host graphs under optimal play.

Contribution

It introduces new results on the game saturation number for multiple graph families and host graphs, including exact formulas and bounds, expanding understanding of saturation games.

Findings

Exact saturation numbers for odd cycles on complete bipartite graphs.

Formula for the saturation game on trees in complete graphs.

Bounds for the saturation game involving paths and bipartite graphs.

Abstract

Given a family ${\mathcal F}$ and a host graph $H$, a graph $G\subseteq H$ is ${\mathcal F}$-saturated relative to $H$ if no subgraph of $G$ lies in ${\mathcal F}$ but adding any edge from $E(H)-E(G)$ to $G$ creates such a subgraph. In the ${\mathcal F}$-saturation game on $H$, players Max and Min alternately add edges of $H$ to $G$, avoiding subgraphs in ${\mathcal F}$, until $G$ becomes ${\mathcal F}$-saturated relative to $H$. They aim to maximize or minimize the length of the game, respectively; $\textrm{sat}_g({\mathcal F};H)$ denotes the length under optimal play (when Max starts). Let ${\mathcal O}$ denote the family of all odd cycles and ${\mathcal T}$ the family of $n$-vertex trees, and write $F$ for ${\mathcal F}$ when ${\mathcal F}=\{F\}$. Our results include $\textrm{sat}_g({\mathcal O};K_{2k})=k^2$, $\textrm{sat}_g({\mathcal T};K_n)=\binom{n-2}{2}+1$ for $n\ge6$, $\textrm{sat}_g(K_{1,3};K_n)=2\lfloor n/2 \rfloor$ for $n\ge8$, $\textrm{sat}_g(K_{1,r+1};K_n)=\frac{rn}{2}-\frac{r^2}{8}+O(1)$, and $|\textrm{sat}_g(P_4;K_n)-(4n-1)/5|\le 1$. We also determine $\textrm{sat}_g(P_4;K_{m,n})$; with $m\ge n$, it is $n$ when $n$ is even, $m$ when $n$ is odd and $m$ is even, and $m+\lfloor n/2 \rfloor$ when $mn$ is odd. Finally, we prove the lower bound $\textrm{sat}_g(C_4;K_{n,n})\ge\frac{1}{10.4}n^{13/12}-O(n^{35/36})$. The results are very similar when Min plays first, except for the $P_4$-saturation game on $K_{m,n}$.