On the path-avoidance vertex-coloring game

For any graph $F$ and any integer $r\geq 2$, the \emph{online vertex-Ramsey density of $F$ and $r$}, denoted $m^*(F,r)$, is a parameter defined via a deterministic two-player Ramsey-type game (Painter vs.\ Builder). This parameter was introduced in a recent paper \cite{mrs11}, where it was shown that the online vertex-Ramsey density determines the threshold of a similar probabilistic one-player game (Painter vs.\ the binomial random graph $G_{n,p}$). For a large class of graphs $F$, including cliques, cycles, complete bipartite graphs, hypercubes, wheels, and stars of arbitrary size, a simple greedy strategy is optimal for Painter and closed formulas for $m^*(F,r)$ are known. In this work we show that for the case where $F=P_\ell$ is a (long) path, the picture is very different. It is not hard to see that $m^*(P_\ell,r)= 1-1/k^*(P_\ell,r)$ for an appropriately defined integer $k^*(P_\ell,r)$, and that the greedy strategy gives a lower bound of $k^*(P_\ell,r)\geq \ell^r$. We construct and analyze Painter strategies that improve on this greedy lower bound by a factor polynomial in $\ell$, and we show that no superpolynomial improvement is possible

On the Lengths of Symmetry Breaking-Preserving Games on Graphs

Given a graph $G$, we consider a game where two players, $A$ and $B$, alternatingly color edges of $G$ in red and in blue respectively. Let $l(G)$ be the maximum number of moves in which $B$ is able to keep the red and the blue subgraphs isomorphic, if $A$ plays optimally to destroy the isomorphism. This value is a lower bound for the duration of any avoidance game on $G$ under the assumption that $B$ plays optimally. We prove that if $G$ is a path or a cycle of odd length $n$, then $\Omega(\log n)\le l(G)\le O(\log^2 n)$. The lower bound is based on relations with Ehrenfeucht games from model theory. We also consider complete graphs and prove that $l(K_n)=O(1)$.Comment: 20 page

Impartial avoidance games for generating finite groups

We study an impartial avoidance game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The first player who cannot select an element without making the set of jointly-selected elements into a generating set for the group loses the game. We develop criteria on the maximal subgroups that determine the nim-numbers of these games and use our criteria to study our game for several families of groups, including nilpotent, sporadic, and symmetric groups.Comment: 14 pages, 4 figures. Revised in response to comments from refere

A Comprehensive Survey of Potential Game Approaches to Wireless Networks

Potential games form a class of non-cooperative games where unilateral improvement dynamics are guaranteed to converge in many practical cases. The potential game approach has been applied to a wide range of wireless network problems, particularly to a variety of channel assignment problems. In this paper, the properties of potential games are introduced, and games in wireless networks that have been proven to be potential games are comprehensively discussed.Comment: 44 pages, 6 figures, to appear in IEICE Transactions on Communications, vol. E98-B, no. 9, Sept. 201

Ramsey games with giants

The classical result in the theory of random graphs, proved by Erdos and Renyi in 1960, concerns the threshold for the appearance of the giant component in the random graph process. We consider a variant of this problem, with a Ramsey flavor. Now, each random edge that arrives in the sequence of rounds must be colored with one of R colors. The goal can be either to create a giant component in every color class, or alternatively, to avoid it in every color. One can analyze the offline or online setting for this problem. In this paper, we consider all these variants and provide nontrivial upper and lower bounds; in certain cases (like online avoidance) the obtained bounds are asymptotically tight.Comment: 29 pages; minor revision