We analyze the following version of the deterministic \hats game. We have a graph $G$, and a sage resides at each vertex of $G$. When the game starts, an adversary puts on the head of each sage a hat of a color arbitrarily chosen from a set of $k$ possible colors. Each sage can see the hat colors of his neighbors but not his own hat color. All of sages are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The strategy is winning if it guarantees at least one correct individual guess for every assignment of colors. Given a graph $G$, its hat guessing number ${\text{HG}}(G)$ is the maximal number $k$ such that there exists a winning strategy.
We disprove the hypothesis that ${\text{HG}}(G) \le Δ+ 1$ and demonstrate that diameter of graph and ${\text{HG}}(G$) are independent.
The following general variant of deterministic Hats game is analyzed. Several sages wearing colored hats occupy the vertices of a graph, the $k$-th sage can have hats of one of $h(k)$ colors. Each sage tries to guess the color of his own hat merely on the basis of observing the hats of his neighbors without exchanging any information. A predetermined guessing strategy is winning if it guarantees at least one correct individual guess for every assignment of colors. For complete graphs and for cycles we solve the problem of describing functions $h(k)$ for which the sages win. We demonstrate here…
We analyze the following general version of the deterministic Hats game. Several sages wearing colored hats occupy the vertices of a graph. Each sage can have a hat of one of $k$ colors. Each sage tries to guess the color of his own hat merely on the basis of observing the hats of his neighbors without exchanging any information. A predetermined guessing strategy is winning if it guarantees at least one correct individual guess for every assignment of colors. We present an example of a planar graph for which the sages win for $k = 14$. We also give an easy proof of the theorem about the Hats g…
We study a cooperative game in which each member of a team of $N$ players, wearing coloured hats and situated at the vertices of a cycle graph $C_N$, is guessing their own hat colour merely on the basis of observing the hats worn by their two neighbours without exchanging the information. Each hat can have one of three colours. A predetermined guessing strategy is winning if it guarantees at least one correct individual guess for every assignment of colours. We prove that a winning strategy exists if and only if $N$ is divisible by $3$ or $N=4$. This problem represents an example of a relation…
We study the hat chromatic number of a graph defined in the following way: there is one player at each vertex of a loopless graph $G$, an adversary places a hat of one of $K$ colors on the head of each player, two players can see each other's hats if and only if they are at adjacent vertices. All players simultaneously try to guess the color of their hat. The players cannot communicate but collectively determine a strategy before the hats are placed. The hat chromatic number, $μ(G)$, is the largest number $K$ of colors such that the players are able to fix a strategy that will ensure that for …
In this note, we give an explicit polynomial-time executable strategy for Peter Winkler's hat guessing game that gives superior results if the distribution of hats is imbalanced. While Winkler's strategy guarantees in any case that $\lfloor n/2 \rfloor$ of the $n$ player guess their hat color correct, our strategy ensures that the players produce $\max\{r,b\} - 1.2 n^{2/3} -2$ correct guesses for any distribution of $r$ red and $b = n - r$ blue hats. We also show that any strategy ensuring $\max\{r,b\} - f(n)$ correct guesses necessarily has $f(n) = Ω(\sqrt n)$.
N players are randomly fitted with a colored hat (q different colors). All players guess simultaneously the color of their own hat observing only the hat colors of the other N-1 players. The team wins if all players guess right. No communication of any sort is allowed, except for an initial strategy session before the game begins. In the first part of our investigation we have q different colors with equal probability. Up to 4 colors we construct optimal strategies for any number of players using Hamming Complete Sets. For 5 colors we find optimal strategies up to 5 players using Optimal Hammi…
The Levine hat game requires $n$ players, each wearing an infinite random stack of black and white hats, to guess the location of a black hat on their own head seeing only the hats worn by all the other players. They are allowed a strategy session before the game, but no further communication. The players collectively win if and only if all their guesses are correct. In this paper we give an overview of what is known about strategies for this game, including an extended discussion of the case with $n = 2$ players (and a conjecture for an optimal strategy in this case). We also prove that $V_n$…
Hat problems have recently become a popular topic in combinatorics and discrete mathematics. These have been shown to be strongly related to coding theory, network coding, and auctions. We consider the following version of the hat game, introduced by Winkler and studied by Butler et al. A team is composed of several players; each player is assigned a hat of a given colour; they do not see their own colour, but can see some other hats, according to a directed graph. The team wins if they have a strategy such that, for any possible assignment of colours to their hats, at least one player guesses…
Several different "hat games" have recently received a fair amount of attention. Typically, in a hat game, one or more players are required to correctly guess their hat colour when given some information about other players' hat colours. Some versions of these games have been motivated by research in complexity theory and have ties to well-known research problems in coding theory, and some variations have led to interesting new research. In this paper, we review Ebert's Hat Game, which garnered a considerable amount of publicity in the late 90's and early 00's, and the Hats-on-a-line Game. The…
A large class of Positional Games are defined on the complete graph on $n$ vertices. The players, Maker and Breaker, take the edges of the graph in turns, and Maker wins iff his subgraph has a given -- usually monotone -- property. Here we introduce the $d$-diameter game, which means that Maker wins iff the diameter of his subgraph is at most $d$. We investigate the biased version of the game; i.e., when the players may take more than one, and not necessarily the same number of edges, in a turn. Our main result is that we proved that the $2$-diameter game has the following surprising property:…
We analyze the version of the deterministic Hats game. In this paper, we present new constructors, i.e. theorems that allow built winning strategies for the sages on different graphs. Using this technique we calculate the hat guessing number ${\text{HG}}_s(G)$ for paths and "petunias", and present a planar graph $G$ for which ${\text{HG}}_1(G) \ge 22$.
Let $γ_g(G)$ be the game domination number of a graph $G$. It is proved that if ${\rm diam}(G) = 2$, then $γ_g(G) \le \left\lceil \frac{n(G)}{2} \right\rceil- \left\lfloor \frac{n(G)}{11}\right\rfloor$. The bound is attained: if ${\rm diam}(G) = 2$ and $n(G) \le 10$, then $γ_g(G) = \left\lceil \frac{n(G)}{2} \right\rceil$ if and only if $G$ is one of seven sporadic graphs with $n(G)\le 6$ or the Petersen graph, and there are exactly ten graphs of diameter $2$ and order $11$ that attain the bound.
We study the localization game on dense random graphs. In this game, a {\em cop} $x$ tries to locate a {\em robber} $y$ by asking for the graph distance of $y$ from every vertex in a sequence of sets $W_1,W_2,\ldots,W_\ell$. We prove high probability upper and lower bounds for the minimum size of each $W_i$ that will guarantee that $x$ will be able to locate $y$.
Let $G$ be a graph with $n$ vertices. The {\em hat guessing number} of $G$ is defined in terms of the following game: There are $n$ players and one opponent. The opponent will wear one of the $q$ hats of different colors on the player's head. At this time, the player can only see the player's hat color at the adjacent vertex, and communication between players is not allowed. Once players are assigned hats, each player must guess the color of his hat at the same time. If at least one player guesses right, then they will win collectively. Given a graph $G$, its hat guessing number $HG(G)$ is the…
The connected domination game was recently introduced by Borowiecki, Fiedorowicz and Sidorowicz as another variation of the domination game. The rules are essentially the same, except that the set of played vertices must be connected at all stages of the game. We answer a problem from their paper regarding the relation between the number of moves in a game where Dominator/Staller starts the game. In this paper we also study the relation to the diameter and present graphs with small connected game domination number. We determine the values on the lexicographic product graphs, and consider the e…
The game of Cops and Robbers is a well known game played on graphs. In this paper we consider the class of graphs of bounded diameter. We improve the strategy of cops and previously used probabilistic method which results in an improved upper bound for the cop number of graphs of bounded diameter. In particular, for graphs of diameter four, we improve the upper bound from $n^{\frac{2}{3}+o(1)}$ to $n^{\frac{3}{5}+o(1)}$ and for diameter three from $n^{\frac{2}{3}+o(1)}$ to $n^{\frac{4}{7}+o(1)}$.
Several variations of hat guessing games have been popularly discussed in recreational mathematics. In a typical hat guessing game, after initially coordinating a strategy, each of $n$ players is assigned a hat from a given color set. Simultaneously, each player tries to guess the color of his/her own hat by looking at colors of hats worn by other players. In this paper, we consider a new variation of this game, in which we require at least $k$ correct guesses and no wrong guess for the players to win the game, but they can choose to "pass". A strategy is called {\em perfect} if it can achieve…
The guessing game introduced by Riis is a variant of the "guessing your own hats" game and can be played on any simple directed graph G on n vertices. For each digraph G, it is proved that there exists a unique guessing number gn(G) associated to the guessing game played on G. When we consider the directed edge to be bidirected, in other words, the graph G is undirected, Christofides and Markstrom introduced a method to bound the value of the guessing number from below using the fractional clique number Kf(G). In particular they showed gn(G) >= |V(G)| - Kf(G). Moreover, it is pointed out that…
We consider the game of Cops and Robber played on the Cartesian product of two trees. Assuming the players play perfectly, it is shown that if there are two cops in the game, then the length of the game (known as the 2-capture time of the graph) is equal to half the diameter of the graph. In particular, the 2-capture time of the m x n grid is proved to be floor ((m+n-2)/2).
A pebbling move refers to the act of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The goal of graph pebbling is: Given an initial distribution of pebbles, use pebbling moves to reach a specified goal vertex called the root. The pebbling number of a graph $π(G)$ is the minimum number of pebbles needed so every distribution of $π(G)$ pebbles can reach every choice of the root. We introduce a new variant of graph pebbling, a game between two players. One player aims to move a pebble to the root and the other player aims to prevent this. We show configurations…
Graph pebbling is a game played on a connected graph G. A player purchases pebbles at a dollar a piece, and hands them to an adversary who distributes them among the vertices of G (called a configuration) and chooses a target vertex r. The player may make a pebbling move by taking two pebbles off of one vertex and moving one pebble to a neighboring vertex. The player wins the game if he can move k pebbles to r. The value of the game (G,k), called the k-pebbling number of G, is the minimum cost to the player to guarantee a win. That is, it is the smallest positive integer m of pebbles so that, …
In this short paper we study the game of Cops and Robbers, played on the vertices of some fixed graph $G$ of order $n$. The minimum number of cops required to capture a robber is called the cop number of $G$. We show that the cop number of graphs of diameter 2 is at most $\sqrt{2n}$, improving a recent result of Lu and Peng by a constant factor. We conjecture that this bound is still not optimal, and obtain some partial results towards the optimal bound.
Pebbling is a game played on a graph. The single player is given a graph and a configuration of pebbles and may make pebbling moves by removing 2 pebbles from one vertex and placing one at an adjacent vertex to eventually have one pebble reach a predetermined vertex. The pebbling number, $π(G)$, is the minimum number of pebbles such that regardless of their exact configuration, the player can use pebbling moves to have a pebble reach any predetermined vertex. Previous work has related $π(G)$ to the diameter of $G$. Clarke, Hochberg, and Hurlbert demonstrated that every connected undirected gra…
We consider the new game of Cops and Attacking Robbers, which is identical to the usual Cops and Robbers game except that if the robber moves to a vertex containing a single cop, then that cop is removed from the game. We study the minimum number of cops needed to capture a robber on a graph $G$, written $cc(G)$. We give bounds on $cc(G)$ in terms of the cop number of $G$ in the classes of bipartite graphs and diameter two, $K_{1,m}$-free graphs.
For $z_1,z_2,z_3 \in \Z^n$, the \emph{tristance} $d_3(z_1,z_2,z_3)$ is a generalization of the $L_1$-distance on $\Z^n$ to a quantity that reflects the relative dispersion of three points rather than two. A tristance anticode $\cA_d$ of diameter $d$ is a subset of $\Z^n$ with the property that $d_3(z_1,z_2,z_3) \leq d$ for all $z_1,z_2,z_3 \in \cA_d$. An anticode is optimal if it has the largest possible cardinality for its diameter $d$. We determine the cardinality and completely classify the optimal tristance anticodes in $\Z^2$ for all diameters $d \ge 1$. We then generalize this result to …
The main topic of this paper is motivated by a localization problem in cellular networks. Given a graph $G$ we want to localize a walking agent by checking his distance to as few vertices as possible. The model we introduce is based on a pursuit graph game that resembles the famous Cops and Robbers game. It can be considered as a game theoretic variant of the \emph{metric dimension} of a graph. We provide upper bounds on the related graph invariant $ζ(G)$, defined as the least number of cops needed to localize the robber on a graph $G$, for several classes of graphs (trees, bipartite graphs, e…
We consider the capacitated selfish replication (CSR) game with binary preferences, over general undirected networks. We first show that such games have an associated ordinary potential function, and hence always admit a pure-strategy Nash equilibrium (NE). Further, when the minimum degree of the network and the number of resources are of the same order, there exists an exact polynomial time algorithm which can find a NE. Following this, we study the price of anarchy of such games, and show that it is bounded above by 3; we further provide some instances for which the price of anarchy is at le…
