Positional games on random graphs

We introduce and study Maker/Breaker-type positional games on random graphs. Our main concern is to determine the threshold probability $p_{F}$ for the existence of Maker's strategy to claim a member of $F$ in the unbiased game played on the edges of random graph $G(n,p)$, for various target families $F$ of winning sets. More generally, for each probability above this threshold we study the smallest bias $b$ such that Maker wins the $(1\:b)$ biased game. We investigate these functions for a number of basic games, like the connectivity game, the perfect matching game, the clique game and the Hamiltonian cycle game

Avoider-Enforcer Game is NP-hard

In an Avoider-Enforcer game, we are given a hypergraph. Avoider and Enforcer alternate in claiming an unclaimed vertex, until all the vertices of the hypergraph are claimed. Enforcer wins if Avoider claims all vertices of an edge; Avoider wins otherwise. We show that it is NP-hard to decide if Avoider has a winning strategy

Annual Report of the Municipal Officers of the Town or Orrington for the Year 1938-1939

https://digitalmaine.com/orrington_books/1094/thumbnail.jp

Games on Graphs

We introduce and study Maker-Breaker positional games on random graphs. Our goal is to determine the threshold probability pF for the existence of Maker's strategy to claim a member of F in the unbiased (one-on-one) game played on the edges of the random graph G(n; p), for various target families F of winning sets. More generally, for each probability above this threshold we study the smallest bias b such that Maker wins the (1: b) biased game. We investigate these functions for a number of basic games, like the connectivity game, the perfect matching game, the clique game, the Hamiltonian cycle game and the tree game. Particular attention is devoted to unbiased games, when b = 1. Next, we consider the planarity game and the k-coloring game on the complete graph on n vertices. In the planarity game the winning sets are all non-planar subgraphs, and in the k-coloring game the winning sets are all non-k-colorable subgraphs. For both of the games we look at a (1: b) biased game. We are interested in determining the largest bias b such that Maker wins the Maker-Breaker version of the game. On the other hand, we want to find the largest bias b such that Forcer wins the Avoider-Forcer version of the game. Finally, we deal with balanced online games on the random graph process. The game is played by a player called Painter. Edges in the random graph process are introduced two at a time. For each pair of edges Painter immediately and irrevocably chooses one of the two possibilities to color one of them red and the other one blue. His goal is to avoid creating a monochromatic copy of a prescribed fixed graph H, for as long as possible. We study the threshold mH for the number of edges to be played to know that Painter almost surely will create a monochromatic copy of H, for H being a cycle, a path and a star

ON THE THRESHOLD FOR THE MAKER-BREAKER H-GAME

We study the Maker-Breaker H-game played on the edge set of the random graph Gn,p. In this game two players, Maker and Breaker, alternately claim unclaimed edges of Gn,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 Gn,p with respect to the graph H

Consistent Digital Line Segments

We introduce a novel and general approach for digitalization of line segments in the plane that satisfies a set of axioms naturally arising from Euclidean axioms. In particular, we show how to derive such a system of digital segments from any total order on the integers. As a consequence, using a well-chosen total order, we manage to define a system of digital segments such that all digital segments are, in Hausdorff metric, optimally close to their corresponding Euclidean segments, thus giving an explicit construction that resolves the main question of Chun et al. (Discrete Comput. Geom. 42(3):359-378, 2009)

On Restricted Min-Wise Independence of Permutations

A family of permutations Sn with a probability distribution on it is called k-restricted min-wise independent if we have Pr[min #(X) = #(x)] = for every subset X |X | # k, every x X , and # chosen at random. We present a simple proof of a result of Norin: every such family has size at least . Some features of our method might be of independent interest