The diameter game

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: Breaker wins the game in which each player chooses one edge per turn, but Maker wins as long as he is permitted to choose $2$ edges in each turn whereas Breaker can choose as many as $(1/9)n^{1/8}/(\ln n)^{3/8}$. In addition, we investigate $d$-diameter games for $d\ge 3$. The diameter games are strongly related to the degree games. Thus, we also provide a generalization of the fair degree game for the biased case.

[1]  András Pluhár.  Generalized Harary Games , 1997, Acta Cybern..

[2]  Tibor Szabó,et al.  Positional games on random graphs , 2006, Random Struct. Algorithms.

[3]  Alfred Lehman,et al.  A Solution of the Shannon Switching Game , 1964 .

[4]  Thomas J. Schaefer,et al.  On the Complexity of Some Two-Person Perfect-Information Games , 1978, J. Comput. Syst. Sci..

[5]  József Beck,et al.  Remarks on positional games. I , 1982 .

[6]  Tomasz Luczak,et al.  Biased Positional Games for Which Random Strategies are Nearly Optimal , 2000, Comb..

[7]  A. Hales,et al.  Regularity and Positional Games , 1963 .

[8]  Tomasz Luczak,et al.  Biased positional games and the phase transition , 2001, Random Struct. Algorithms.

[9]  József Beck,et al.  Random Graphs and Positional Games on the Complete Graph , 1985 .

[10]  József Beck,et al.  Deterministic Graph Games and a Probabilistic Intuition , 1994, Combinatorics, Probability and Computing.

[11]  Michael Krivelevich,et al.  Planarity, Colorability, and Minor Games , 2008, SIAM J. Discret. Math..

[12]  P. Erdös,et al.  Biased Positional Games , 1978 .

[13]  András Pluhár.  The Recycled Kaplansky's Game , 2004, Acta Cybern..

[14]  Paul Erdös,et al.  On a Combinatorial Game , 1973, J. Comb. Theory A.

[15]  András Pluhár.  The accelerated k-in-a-row game , 2002, Theor. Comput. Sci..

[16]  Béla Bollobás,et al.  Random Graphs, Second Edition , 2001, Cambridge Studies in Advanced Mathematics.

[17]  B. Bollobás,et al.  Combinatorics, Probability and Computing , 2006 .

[18]  Nándor Sieben.  Hexagonal polyomino weak (1, 2)-achievement games , 2004, Acta Cybern..