Positional games on randomly perturbed graphs

TL;DR

This paper investigates Maker-Breaker positional games on graphs formed by combining a dense deterministic graph with a random graph, establishing thresholds for winning conditions like Hamiltonicity and connectivity.

Contribution

It introduces the study of Maker-Breaker games on randomly perturbed graphs and determines threshold probabilities for key winning properties based on graph parameters.

Findings

Thresholds for Hamiltonicity game on perturbed graphs

Thresholds for k-connectivity game on perturbed graphs

Optimal results for Waiter-Client versions of these games

Abstract

Maker-Breaker games are played on a hypergraph $(X,\mathcal{F})$, where $\mathcal{F} \subseteq 2^X$ denotes the family of winning sets. Both players alternately claim a predefined amount of edges (called bias) from the board $X$, and Maker wins the game if she is able to occupy any winning set $F \in \mathcal{F}$. These games are well studied when played on the complete graph $K_n$ or on a random graph $G_{n,p}$. In this paper we consider Maker-Breaker games played on randomly perturbed graphs instead. These graphs consist of the union of a deterministic graph $G_\alpha$ with minimum degree at least $\alpha n$ and a binomial random graph $G_{n,p}$. Depending on $\alpha$ and Breaker's bias $b$ we determine the order of the threshold probability for winning the Hamiltonicity game and the $k$-connectivity game on $G_{\alpha}\cup G_{n,p}$, and we discuss the $H$-game when $b=1$. Furthermore, we give optimal results for the Waiter-Client versions of all mentioned games.