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By now, the Maker-Breaker connectivity game on a complete graph $K_n$ or on a random graph $Gsim G_{n,p}$ is well studied. Recently, London and Pluhar 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 $Gsim G_{n,p}$ for winning the game drastically differ from the corresponding values for the usual Maker-Breaker version, assuming Makers bias to be $1$. However, they observed that the threshold biases of bo
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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 claim all edges of a sufficiently colourful infinite complete subgraph and Breakers aim being to prevent this. We show that if there are only finitely many colours then Maker can obtain a complete subgraph in which all colours appear infinitely often, but that Breaker can prevent this if there are infinitely many colours. Even when there are infinitely many colours, we show that Maker can obtain a complete subgraph in which infinitely many of the colours each appear infinitely often.
Number the cells of a (possibly infinite) chessboard in some way with the numbers 0, 1, 2, ... Consider the cells in order, placing a queen in a cell if and only if it would not attack any earlier queen. The problem is to determine the positions of the queens. We study the problem for a doubly-infinite chessboard of size Z x Z numbered along a square spiral, and an infinite single-quadrant chessboard (of size N x N) numbered along antidiagonals. We give a fairly complete solution in the first case, based on the Tribonacci word. There are connections with combinatorial games.
We study a game where two players take turns selecting points of a convex geometry until the convex closure of the jointly selected points contains all the points of a given winning set. The winner of the game is the last player able to move. We develop a structure theory for these games and use it to determine the nim number for several classes of convex geometries, including one-dimensional affine geometries, vertex geometries of trees, and games with a winning set consisting of extreme points.
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 Breakers 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-Clie
We consider the Maker-Breaker positional game on the vertices of the $n$-dimensional hypercube ${0,1}^n$ with $k$-dimensional subcubes as winning sets. We describe a pairing strategy which allows Breaker to win when $k = n/4 +1$ in the case where $n$ is a power of 4. Our results also imply the general result that there is a Breakers win pairing strategy for any $n geq 3$ if $k = leftlfloorfrac{3}{7}nrightrfloor +1$.
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