On Avoider-Enforcer games

TL;DR

This paper analyzes the Avoider-Enforcer game on complete graphs, determining the minimal number of rounds Enforcer needs to guarantee a certain subgraph property, using advanced combinatorial tools like the Szemerédi Regularity Lemma.

Contribution

It provides asymptotic values for the parameter _E(\u210f) for families of graphs with and without bipartite members, extending understanding of Avoider-Enforcer game dynamics.

Findings

_E(\u210f) is asymptotically determined for non-bipartite graph families.

_E() = o(n^2) if the family contains a bipartite graph.

The proof employs the game of JumbleG and Szemerédi Regularity Lemma.

Abstract

In the Avoider-Enforcer game on the complete graph $K_n$, the players (Avoider and Enforcer) each take an edge in turn. Given a graph property $\mathcal{P}$, Enforcer wins the game if Avoider's graph has the property $\mathcal{P}$. An important parameter is $\tau_E({\cal P})$, the smallest integer $t$ such that Enforcer can win the game against any opponent in $t$ rounds. In this paper, let $\mathcal{F}$ be an arbitrary family of graphs and $\mathcal{P}$ be the property that a member of $\mathcal{F}$ is a subgraph or is an induced subgraph. We determine the asymptotic value of $\tau_E(\mathcal{P})$ when $\mathcal{F}$ contains no bipartite graph and establish that $\tau_E(\mathcal{P})=o(n^2)$ if $\mathcal{F}$ contains a bipartite graph. The proof uses the game of JumbleG and the Szemer\'edi Regularity Lemma.