Related papers: On Balanced Coloring Games in Random Graphs
When many colors appear in edge-colored graphs, it is only natural to expect rainbow subgraphs to appear. This anti-Ramsey problem has been studied thoroughly and yet there remain many gaps in the literature. Expanding upon classical and…
Game balancing is an important part of the (computer) game design process, in which designers adapt a game prototype so that the resulting gameplay is as entertaining as possible. In industry, the evaluation of a game is often based on…
The strong Ramsey game $R(\mathcal{B}, H)$ is a two-player game played on a graph $\mathcal{B}$, referred to as the board, with a target graph $H$. In this game, two players, $P_1$ and $P_2$, alternately claim unclaimed edges of…
Given two graphs $G$ and $H$, we investigate for which functions $p=p(n)$ the random graph $G_{n,p}$ (the binomial random graph on $n$ vertices with edge probability $p$) satisfies with probability $1-o(1)$ that every red-blue-coloring of…
A homomorphism from a graph $X$ to a graph $Y$ is an adjacency preserving mapping $f:V(X) \rightarrow V(Y)$. We consider a nonlocal game in which Alice and Bob are trying to convince a verifier with certainty that a graph $X$ admits a…
For two graphs, $G$ and $F$, and an integer $r\ge2$ we write $G\rightarrow (F)_r$ if every $r$-coloring of the edges of $G$ results in a monochromatic copy of $F$. In 1995, the first two authors established a threshold edge probability for…
Given two graphs $H_1$ and $H_2$, an online Ramsey game is played on the edge set of $K_\mathbb{N}$. In every round Builder selects an edge and Painter colors it red or blue. Builder is trying to force Painter to create a red copy of $H_1$…
We introduce a way to parameterize automata and games on finite graphs with natural numbers. The parameters are accessed essentially by allowing counting down from the parameter value to 0 and branching depending on whether 0 has been…
In this paper we study a type of games regularized by the relative entropy, where the players' strategies are coupled through a random environment variable. Besides the existence and the uniqueness of equilibria of such games, we prove that…
One of the natural objectives of the field of the social networks is to predict agents' behaviour. To better understand the spread of various products through a social network arXiv:1105.2434 introduced a threshold model, in which the nodes…
The Strong Ramsey game $\mathcal{R}(B,G)$ is a two player game with players $P_1$ and $P_2$, where $B$ and $G$ are $k$-uniform hypergraphs for some $k \geq 2$. $G$ is always finite, while $B$ may be infinite. $P_1$ and $P_2$ alternately…
In the game of Matching Pennies, Alice and Bob each hold a penny, and at every tick of the clock they simultaneously display the head or the tail sides of their coins. If they both display the same side, then Alice wins Bob's penny; if they…
Consider the following game. We are given a tree $T$ and two players (say) Alice and Bob who alternately colour an edge of a tree (using one of $k$ colours). If all edges of the tree get coloured, then Alice wins else Bob wins. Game…
A graph $G$ is Ramsey for a graph $H$ if every 2-colouring of the edges of $G$ contains a monochromatic copy of $H$. We consider the following question: if $H$ has bounded treewidth, is there a `sparse' graph $G$ that is Ramsey for $H$? Two…
We introduce a new abstract graph game, Swap Planarity, where the goal is to reach a state without edge intersections and a move consists of swapping the locations of two vertices connected by an edge. We analyze this puzzle game using…
We consider a simple game, the $k$-regular graph game, in which players take turns adding edges to an initially empty graph subject to the constraint that the degrees of vertices cannot exceed $k$. We show a sharp topological threshold for…
The online semi-random graph process is a one-player game which starts with the empty graph on $n$ vertices. At every round, a player (called Builder) is presented with a vertex $v$ chosen uniformly at random and independently from previous…
We consider the single-conflict coloring problem, a graph coloring problem in which each edge of a graph receives a forbidden ordered color pair. The task is to find a vertex coloring such that no two adjacent vertices receive a pair of…
An $n$-vertex graph is called $C$-Ramsey if it has no clique or independent set of size $C\log_2 n$ (i.e., if it has near-optimal Ramsey behavior). In this paper, we study edge-statistics in Ramsey graphs, in particular obtaining very…
We study biased Maker-Breaker positional games between two players, one of whom is playing randomly against an opponent with an optimal strategy. In this paper we consider the scenario when Maker plays randomly and Breaker is "clever", and…