We investigate Maker-Breaker games on graphs of size $\aleph_1$ in which Maker's goal is to build a copy of the host graph. We establish a firm dependence of the outcome of the game on the axiomatic framework. Relating to this, we prove that there is a winning strategy for Maker in the $K_{ω,ω_1}$-game under ZFC+MA+$\neg$CH and a winning strategy for Breaker under ZFC+CH. We prove a similar result for the $K_{ω_1}$-game. Here, Maker has a winning strategy under ZF+DC+AD, while Breaker has one under ZFC+CH again.
In this paper we study the (a : b) Maker-Breaker Connectivity game, played on the edge-set of the complete graph on n vertices. We determine the winner for almost all values of a and b.
We study the Maker-Breaker $H$-game played on the edge set of the random graph $G_{n,p}$. In this game two players, Maker and Breaker, alternately claim unclaimed edges of $G_{n,p}$, until all the edges are claimed. Maker wins if he claims all the edges of a copy of a fixed graph $H$; Breaker wins otherwise. In this paper we show that, with the exception of trees and triangles, the threshold for an $H$-game is given by the threshold of the corresponding Ramsey property of $G_{n,p}$ with respect to the graph $H$.
We study two biassed Maker-Breaker games played on the complete digraph $\vec{K}_n$. In the strong connectivity game, Maker wants to build a strongly connected subgraph. We determine the asymptotic optimal bias for this game viz. $\frac{n}{\log n}$. In the Hamiltonian game, Maker wants to build a Hamiltonian subgraph. We determine the asymptotic optimal bias for this game up to a constant factor.
We study the $(p,q)$-Maker Breaker Crossing game introduced by Day and Falgas Ravry in 'Maker-Breaker percolation games I: crossing grids'. The game described in their paper involves two players Maker and Breaker who take turns claiming p and q as yet unclaimed edges of the graph respectively. Maker aims to make a horizontal path from a leftmost vertex to a rightmost vertex and Breaker aims to prevent this. The game is a version of the more general Shannon switching game and is played on a square grid graph. We consider the same game played on the triangular grid graph $Δ_{(m,n)}$ (m vertices …
For a tree T on n vertices, we study the Maker-Breaker game, played on the edge set of the complete graph on n vertices, which Maker wins as soon as the graph she builds contains a copy of T. We prove that if T has bounded maximum degree, then Maker can win this game within n+1 moves. Moreover, we prove that Maker can build almost every tree on n vertices in n-1 moves and provide non-trivial examples of families of trees which Maker cannot build in n-1 moves.
Given an integer-valued matrix $A$ of dimension $\ell \times k$ and an integer-valued vector $b$ of dimension $\ell$, the Maker-Breaker $(A,b)$-game on a set of integers $X$ is the game where Maker and Breaker take turns claiming previously unclaimed integers from $X$, and Maker's aim is to obtain a solution to the system $Ax=b$, whereas Breaker's aim is to prevent this. When $X$ is a random subset of $\{1,\dots,n\}$ where each number is included with probability $p$ independently of all others, we determine the threshold probability $p_0$ for when the game is Maker or Breaker's win, for a lar…
A rank-3 Maker-Breaker game is played on a hypergraph in which all hyperedges are sets of at most 3 vertices. The two players of the game, called Maker and Breaker, move alternately. On his turn, maker chooses a vertex to be withdrawn from all hyperedges, while Breaker on her turn chooses a vertex and delete all the hyperedges containing that vertex. Maker wins when by the end of his turn some hyperedge is completely covered, i.e. the last remaining vertex of that hyperedge is withdrawn. Breaker wins when by the end of her turn, all hyperedges have been deleted. Solving a Maker-Breaker game is…
In this paper we analyze classical Maker-Breaker games played on the edge set of a sparse random board $G\sim \gnp$. We consider the Hamiltonicity game, the perfect matching game and the $k$-connectivity game. We prove that for $p(n)\geq \text{polylog}(n)/n$, the board $G\sim \gnp$ is typically such that Maker can win these games asymptotically as fast as possible, i.e. within $n+o(n)$, $n/2+o(n)$ and $kn/2+o(n)$ moves respectively.
In a $(1:b)$ biased Maker-Breaker game, how good a strategy is for a player can be measured by the bias range for which its rival can win, choosing an appropriate counterstrategy. Bednarska and Łuczak proved that, in the $H$-subgraph game, the uniformly random strategy for Maker is essentially optimal with high probability. Here we prove an analogous result for the $H$-graph minor game, and we study for which choices of $H$ the random strategy is within a factor of $1+o(1)$ of being optimal.
We study the $(m,b)$ Maker-Breaker percolation game on $\mathbb{Z}^2$, introduced by Day and Falgas-Ravry. As our first result, we show that Breaker has a winning strategy for the $(m,b)$-game whenever $b \geq (2-\frac{1}{14} + o(1))m$, breaking the ratio $2$ barrier proved by Day and Falgas-Ravry. Addressing further questions of Day and Falgas-Ravry, we show that Breaker can win the $(m,2m)$-game even if he allows Maker to claim $c$ edges before the game starts, for any integer $c$, and that he can moreover win rather fast (as a function of $c$). Finally, we consider the game played on $\math…
Maker-Breaker total domination game in graphs is introduced as a natural counterpart to the Maker-Breaker domination game recently studied by Duchêne, Gledel, Parreau, and Renault. Both games are instances of the combinatorial Maker-Breaker games. The Maker-Breaker total domination game is played on a graph $G$ by two players who alternately take turns choosing vertices of $G$. The first player, Dominator, selects a vertex in order to totally dominate $G$ while the other player, Staller, forbids a vertex to Dominator in order to prevent him to reach his goal. It is shown that there are infinit…
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. In particular, we show that for $b=o\left(\sqrt{n}\right)$, Maker can build a pancyclic graph (that is, a graph that contains cycles of every possible length) while playing a $(1:b)$ game on $E(K_n)$. As another application, we show that for $b=Θ\left(n/\ln n\right)$, playing a $(1:b)$ game on $E(K_n)$, Maker can build a graph which contains copies of all sp…
In a Maker-Breaker game there are two players, Maker and Breaker, where Maker wins if they create a specified structure while Breaker wins if they prevent Maker from winning indefinitely. A $3$-term arithmetic progression, or $3$-AP, is a sequence of three distinct integers $a, b, c$ such that $b-a = c-b$. The $3$-AP game is a biased Maker-Breaker game played on $[n]$ where every round Breaker selects $q$ unclaimed integers for every Maker's one integer. Maker is trying to select points such that they have a $3$-AP and Breaker is trying to prevent this. The main question of interest is determi…
In a $(1:b)$ Maker-Breaker game, a primary question is to find the maximal value of $b$ that allows Maker to win the game (that is, the critical bias $b^*$). Erdős conjectured that the critical bias for many Maker-Breaker games played on the edge set of $K_n$ is the same as if both players claim edges randomly. Indeed, in many Maker-Breaker games, "Erdős Paradigm" turned out to be true. Therefore, the next natural question to ask is the (typical) value of the critical bias for Maker-Breaker games where only one player claims edges randomly. A random-player Maker-Breaker game is a two-player ga…
The triangle game introduced by Chvátal and Erdős (1978) is one of the most famous combinatorial games. For $n,q\in\mathbb{N}$, the $(n,q)$-triangle game is played by two players, called Maker and Breaker, on the complete graph $K_n$. Alternately Maker claims one edge and thereafter Breaker claims $q$ edges of the graph. Maker wins the game if he can claim all three edges of a triangle, otherwise Breaker wins. Chvátal and Erdős (1978) proved that for $q<\sqrt{2n+2}-5/2\approx 1.414\sqrt{n}$ Maker has a winning strategy, and for $q\geq 2\sqrt{n}$ Breaker has a winning strategy. Since then, the …
Given a tree $T=(V,E)$ on $n$ vertices, we consider the $(1 : q)$ Maker-Breaker tree embedding game ${\mathcal T}_n$. The board of this game is the edge set of the complete graph on $n$ vertices. Maker wins ${\mathcal T}_n$ if and only if he is able to claim all edges of a copy of $T$. We prove that there exist real numbers $α, ε> 0$ such that, for sufficiently large $n$ and for every tree $T$ on $n$ vertices with maximum degree at most $n^ε$, Maker has a winning strategy for the $(1 : q)$ game ${\mathcal T}_n$, for every $q \leq n^α$. Moreover, we prove that Maker can win this game within $n …
We introduce and analyze the Walker-Breaker game, a variant of Maker-Breaker games where Maker is constrained to choose edges of a walk or path in a given graph G, with the goal of visiting as many vertices of the underlying graph as possible.
In this paper, we study Maker-Breaker games on the random hypergraph $H_{n,s,p}$, obtained from the complete $s$-graph by keeping every edge independently with probability $p$. We determine the threshold probability for the property of Maker winning the game as a function of $s$, the uniformity of the underlying hypergraph, as well as $m$, $b$, the number of vertices that Maker and Breaker are respectively allowed to pick each turn. In addition, we show that depending on those $m,b,s$, there are two types of thresholds: either being Maker-win is a local property and the threshold is weak, or i…
We introduce the Maker-Breaker domination game, a two player game on a graph. At his turn, the first player, Dominator, select a vertex in order to dominate the graph while the other player, Staller, forbids a vertex to Dominator in order to prevent him to reach his goal. Both players play alternately without missing their turn. This game is a particular instance of the so-called Maker-Breaker games, that is studied here in a combinatorial context. In this paper, we first prove that deciding the winner of the Maker-Breaker domination game is PSPACE-complete, even for bipartite graphs and split…
Let $Λ$ be an infinite connected graph, and let $v_0$ be a vertex of $Λ$. We consider the following positional game. Two players, Maker and Breaker, play in alternating turns. Initially all edges of $Λ$ are marked as unsafe. On each of her turns, Maker marks $p$ unsafe edges as safe, while on each of his turns Breaker takes $q$ unsafe edges and deletes them from the graph. Breaker wins if at any time in the game the component containing $v_0$ becomes finite. Otherwise if Maker is able to ensure that $v_0$ remains in an infinite component indefinitely, then we say she has a winning strategy. Th…
Charles University, Czech Republic（捷克查理大学）；University of Novi Sad, Serbia（塞尔维亚诺维萨德大学）；ELTE Eötvös Loránd University（匈牙利罗兰大学）；University College London, UK（英国伦敦大学学院）；Czech Technical University in Prague, Czech Republic（捷克布拉格工业大学）；University of Auckland, New Zealand（新西兰奥克兰大学）
We consider a Maker-Breaker type game on the plane, in which each player takes $t$ points on their $t^\textrm{th}$ turn. Maker wins if he obtains $n$ points on a line (in any direction) without any of Breaker's points between them. We show that, despite Maker's apparent advantage, Breaker can prevent Maker from winning until about his $n^\textrm{th}$ turn. We actually prove a stronger result: that Breaker only needs to play $ω(\log t)$ points on his $t^\textrm{th}$ turn to prevent Maker from winning until this time. We also consider the situation when the number of points claimed by Maker grow…
In a Maker-Breaker game on a graph $G$, Breaker and Maker alternately claim edges of $G$. Maker wins if, after all edges have been claimed, the graph induced by his edges has some desired property. We consider four Maker-Breaker games played on random geometric graphs. For each of our four games we show that if we add edges between $n$ points chosen uniformly at random in the unit square by order of increasing edge-length then, with probability tending to one as $n\to\infty$, the graph becomes Maker-win the very moment it satisfies a simple necessary condition. In particular, with high probabi…
Motivated by problems in percolation theory, we study the following 2-player positional game. Let $Λ_{m \times n}$ be a rectangular grid-graph with $m$ vertices in each row and $n$ vertices in each column. Two players, Maker and Breaker, play in alternating turns. On each of her turns, Maker claims $p$ (as-yet unclaimed) edges of the board $Λ_{m \times n}$, while on each of his turns Breaker claims $q$ (as-yet unclaimed) edges of the board and destroys them. Maker wins the game if she manages to claim all the edges of a crossing path joining the left-hand side of the board to its right-hand si…
Let $d(x,y)$ denote the length of a shortest path between vertices $x$ and $y$ in a graph $G$ with vertex set $V$. For a positive integer $k$, let $d_k(x,y)=\min\{d(x,y), k+1\}$ and $R_k\{x,y\}=\{z\in V: d_k(x,z) \neq d_k(y,z)\}$. A set $S \subseteq V$ is a \emph{distance-$k$ resolving set} of $G$ if $S \cap R_k\{x,y\} \neq\emptyset$ for distinct $x,y\in V$. In this paper, we study the maker-breaker distance-$k$ resolving game (MB$k$RG) played on a graph $G$ by two players, Maker and Breaker, who alternately select a vertex of $G$ not yet chosen. Maker wins by selecting vertices which form a d…
In this paper, we construct two hypergraphs which exhibit the following properties. We first construct a hypergraph $G_{CP}$ and show that Breaker wins the Maker-Breaker game on $G_{CP}$, but Chooser wins the Chooser-Picker game on $G_{CP}$. This disproves an (informally stated) conjecture of Beck. Our second construction relates to Beck's Neighbourhood Conjecture, which (in its weakest form) states that there exists $c > 1$ such that Breaker wins the Maker-Breaker game on any $n$-uniform hypergraph $G$ of maximum degree at most $c^n$. We consider the case n=4 and construct a 4-graph $G_4$ wit…
We study Maker/Breaker games on the edges of sparse graphs. Maker and Breaker take turns in claiming previously unclaimed edges of a given graph H. Maker aims to occupy a given target graph G and Breaker tries to prevent Maker from achieving his goal. We define a function f and show that for every d-regular graph G on n vertices there is a graph H with at most f(d)n edges such that Maker can occupy a copy of G in the game on H.
Let $G$ be a complete convex geometric graph whose vertex set $P$ forms a convex polygon $C$, and let $F$ be a family of subgraphs of $G$. A blocker for $F$ is a set of edges, of smallest possible size, that contains a common edge with every element of $F$. Previous works determined the blockers for various families $F$ of non-crossing subgraphs, including the families of all perfect matchings, all spanning trees, all Hamiltonian paths, etc. In this paper we present a complete characterization of the family $B$ of blockers for the family $T$ of triangulations of $C$. In particular, we show tha…
We look at the unbiased Maker-Breaker Hamiltonicity game played on the edge set of a complete graph $K_n$, where Maker's goal is to claim a Hamiltonian cycle. First, we prove that, independent of who starts, Maker can win the game for $n = 8$ and $n = 9$. Then we use an inductive argument to show that, independent of who starts, Maker can win the game if and only if $n \geq 8$. This, in particular, resolves in the affirmative the long-standing conjecture of Papaioannou. We also study two standard positional games related to Hamiltonicity game. For Hamiltonian Path game, we show that Maker can …
We consider random-turn positional games, introduced by Peres, Schramm, Sheffield and Wilson in 2007. A $p$-random-turn positional game is a two-player game, played the same as an ordinary positional game, except that instead of alternating turns, a coin is being tossed before each turn to decide the identity of the next player to move (the probability of Player I to move is $p$). We analyze the random-turn version of several classical Maker-Breaker games such as the game Box (introduced by Chvátal and Erd\H os in 1987), the Hamilton cycle game and the $k$-vertex-connectivity game (both played…
In a biased weak $(a,b)$ polyform achievement game, the maker and the breaker alternately mark $a,b$ previously unmarked cells on an infinite board, respectively. The maker's goal is to mark a set of cells congruent to a polyform. The breaker tries to prevent the maker from achieving this goal. A winning maker strategy for the $(a,b)$ game can be built from winning strategies for games involving fewer marks for the maker and the breaker. A new type of breaker strategy called the priority strategy is introduced. The winners are determined for all $(a,b)$ pairs for polyiamonds and polyominoes up…
By now, the Maker-Breaker connectivity game on a complete graph $K_n$ or on a random graph $G\sim G_{n,p}$ is well studied. Recently, London and Pluhár suggested a variant in which Maker always needs to choose her edges in such a way that her graph stays connected. By their results it follows that for this connected version of the game, the threshold bias on $K_n$ and the threshold probability on $G\sim G_{n,p}$ for winning the game drastically differ from the corresponding values for the usual Maker-Breaker version, assuming Maker's bias to be $1$. However, they observed that the threshold bi…
In this paper we analyze biased Maker-Breaker games and Avoider-Enforcer games, both played on the edge set of a random board $G\sim \gnp$. In Maker-Breaker games there are two players, denoted by Maker and Breaker. In each round, Maker claims one previously unclaimed edge of $G$ and Breaker responds by claiming $b$ previously unclaimed edges. We consider the Hamiltonicity game, the perfect matching game and the $k$-vertex-connectivity game, where Maker's goal is to build a graph which possesses the relevant property. Avoider-Enforcer games are the reverse analogue of Maker-Breaker games with …
The Maker-Breaker domination game is played on a graph $G$ by Dominator and Staller. The players alternatively select a vertex of $G$ that was not yet chosen in the course of the game. Dominator wins if at some point the vertices he has chosen form a dominating set. Staller wins if Dominator cannot form a dominating set. In this paper we introduce the Maker-Breaker domination number $γ_{\rm MB}(G)$ of $G$ as the minimum number of moves of Dominator to win the game provided that he has a winning strategy and is the first to play. If Staller plays first, then the corresponding invariant is denot…
We study Maker-Breaker games played on the edge set of a random graph. Specifically, we consider the random graph process and analyze the first time in a typical random graph process that Maker starts having a winning strategy for his final graph to admit some property $\mP$. We focus on three natural properties for Maker's graph, namely being $k$-vertex-connected, admitting a perfect matching, and being Hamiltonian. We prove the following optimal hitting time results: with high probability Maker wins the $k$-vertex connectivity game exactly at the time the random graph process first reaches m…
Positional games have been introduced by Hales and Jewett in 1963 and have been extensively investigated in the literature since then. These games are played on a hypergraph where two players alternately select an unclaimed vertex of it. In the Maker-Breaker convention, if Maker manages to fully take a hyperedge, she wins, otherwise, Breaker is the winner. In the Maker-Maker convention, the first player to take a hyperedge wins. In both cases, the game stops as soon as Maker has taken a hyperedge. By definition, this family of games does not handle scores and cannot represent games in which pl…
