On winning fast in Avoider-Enforcer games

TL;DR

This paper investigates the maximum duration of Avoider-Enforcer games on complete graphs for three graph properties, providing strategies for Avoider to prolong the game close to extremal bounds.

Contribution

It establishes near-optimal lower bounds for Avoider's moves in three different positional games, matching known upper bounds up to a small additive constant.

Findings

Avoider can keep the graph outerplanar for at least 2n-8 moves.

Avoider can play for at least d(n)-3 moves in the diamond-free game.

Avoider can maintain a k-degenerate graph for e(n) moves, close to the maximum edges.

Abstract

We analyze the duration of the unbiased Avoider-Enforcer game for three basic positional games. All the games are played on the edges of the complete graph on $n$ vertices, and Avoider's goal is to keep his graph outerplanar, diamond-free and $k$-degenerate, respectively. It is clear that all three games are Enforcer's wins, and our main interest lies in determining the largest number of moves Avoider can play before losing. Extremal graph theory offers a general upper bound for the number of Avoider's moves. As it turns out, for all three games we manage to obtain a lower bound that is just an additive constant away from that upper bound. In particular, we exhibit a strategy for Avoider to keep his graph outerplanar for at least $2n-8$ moves, being just 6 short of the maximum possible. A diamond-free graph can have at most $d(n)=\lceil\frac{3n-5}{2}\rceil$ edges, and we prove that Avoider can play for at least $d(n)-3$ moves. Finally, if $k$ is small compared to $n$, we show that Avoider can keep his graph $k$-degenerate for as many as $e(n)$ moves, where $e(n)$ is the maximum number of edges a $k$-degenerate graph can have.