Related papers: Hitting time results for Maker-Breaker games
In the Maker-Breaker vertex colouring game, first publicised by Gardner in 1981, Maker and Breaker alternately colour vertices of a graph using a fixed palette, maintaining a proper colouring at all times. Maker aims to colour the whole…
We investigate a game played between two players, Maker and Breaker, on a countably infinite complete graph where the vertices are the rational numbers. The players alternately claim unclaimed edges. It is Maker's goal to have after…
In the $\left(1:b\right)$ component game played on a graph $G$, two players, Maker and Breaker, alternately claim~$1$ and~$b$ previously unclaimed edges of $G$, respectively. Maker's aim is to maximise the size of a largest connected…
We consider a random geometric graph process where random points $(X_i)_{i \ge 1}$ are embedded consecutively in the $d$-dimensional unit torus $\mathbb{T}^d$, and every two points at distance at most $r$ form an edge. As $r\to 0$, we…
Waiter-Client games are played on some hypergraph $(X,\mathcal{F})$, where $\mathcal{F}$ denotes the family of winning sets. For some bias $b$, during each round of such a game Waiter offers to Client $b+1$ elements of $X$, of which Client…
Consider the random subgraph process on a base graph $G$ on $n$ vertices: a sequence $\lbrace G_t \rbrace _{t=0} ^{|E(G)|}$ of random subgraphs of $G$ obtained by choosing an ordering of the edges of $G$ uniformly at random, and by…
The Maker-Breaker domination game (MBD game) is a two-player game played on a graph $G$ by Dominator and Staller. They alternately select unplayed vertices of $G$. The goal of Dominator is to form a dominating set with the set of vertices…
We introduce and study a Maker-Breaker type game in which the issue is to create or avoid two disjoint dominating sets in graphs without isolated vertices. We prove that the maker has a winning strategy on all connected graphs if the game…
We study the Maker-Breaker domination game played by Dominator and Staller on the vertex set of a given graph. Dominator wins when the vertices he has claimed form a dominating set of the graph. Staller wins if she makes it impossible for…
We study Hamiltonicity in the union of an $n$-vertex graph $H$ with high minimum degree and a binomial random graph on the same vertex set. In particular, we consider the case when $H$ has minimum degree close to $n/2$. We determine the…
We consider the following definition of connectivity in $k$-uniform hypergraphs: Two $j$-sets are $j$-connected if there is a walk of edges between them such that two consecutive edges intersect in at least $j$ vertices. We determine the…
The $(m,b)$ Maker-Breaker percolation game on $(\mathbb{Z}^2)_p$, introduced by Day and Falgas-Ravry, is played in the following way. Before the game starts, each edge of $\mathbb{Z}^2$ is removed independently with probability $1-p$. After…
We present a general approach connecting biased Maker-Breaker games and problems about local resilience in random graphs. We utilize this approach to prove new results and also to derive some known results about biased Maker-Breaker games.…
The Maker-Breaker domination game is played on a graph $G$ by two players, called Dominator and Staller, who alternately choose a vertex that has not been played so far. Dominator wins the game if his moves form a dominating set. Staller…
We consider the following budget-constrained random graph process introduced by Frieze, Krivelevich and Michaeli. A player, called Builder, is presented with $t$ distinct edges of $K_n$ one by one, chosen uniformly at random. Builder may…
We investigate games played between Maker and Breaker on an infinite complete graph whose vertices are coloured with colours from a given set, each colour appearing infinitely often. The players alternately claim edges, Makers aim being to…
Let $r \ge 4$ be an integer and consider the following game on the complete graph $K_n$ for $n \in r \mathbb{Z}$: Two players, Maker and Breaker, alternately claim previously unclaimed edges of $K_n$ such that in each turn Maker claims one…
We study the (1:b) Maker-Breaker component game, played on the edge set of a d-regular graph. Maker's aim in this game is to build a large connected component, while Breaker's aim is to not let him do so. For all values of Breaker's bias b,…
We initiate the study of a new variant of the Maker-Breaker positional game, which we call multistage game. Given a hypergraph $\mathcal{H}=(\mathcal{X},\mathcal{F})$ and a bias $b \ge 1$, the $(1:b)$ multistage Maker-Breaker game on…
The semi-random graph process is an adaptive random graph process in which an online algorithm is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the algorithm independently and uniformly at…