Avoider-Enforcer star games

TL;DR

This paper analyzes Avoider-Enforcer star games on complete graphs, providing explicit strategies and threshold biases for both strict and monotone versions, advancing understanding of positional game thresholds.

Contribution

It introduces explicit strategies and determines threshold biases for Avoider-Enforcer star games, covering both strict and monotone variants, for all constant k ≥ 3.

Findings

Explicit winning strategies for both players in strict and monotone versions.

Order of magnitude of threshold biases for the game.

Analysis valid for all constant k ≥ 3.

Abstract

In this paper, we study $(1 : b)$ Avoider-Enforcer games played on the edge set of the complete graph on $n$ vertices. For every constant $k\geq 3$ we analyse the $k$-star game, where Avoider tries to avoid claiming $k$ edges incident to the same vertex. We analyse both versions of Avoider-Enforcer games -- the strict and the monotone -- and for each provide explicit winning strategies for both players. We determine the order of magnitude of the threshold biases $f^{mon}_\mathcal{F}$, $f^-_\mathcal{F}$ and $f^+_\mathcal{F}$, where $\mathcal{F}$ is the hypergraph of the game.