284 research outputs found
Hitting time results for Maker-Breaker games
We study Maker-Breaker games played on the edge set of a random graph.
Specifically, we consider the random graph process and analyze the first time
in a typical random graph process that Maker starts having a winning strategy
for his final graph to admit some property \mP. We focus on three natural
properties for Maker's graph, namely being -vertex-connected, admitting a
perfect matching, and being Hamiltonian. We prove the following optimal hitting
time results: with high probability Maker wins the -vertex connectivity game
exactly at the time the random graph process first reaches minimum degree ;
with high probability Maker wins the perfect matching game exactly at the time
the random graph process first reaches minimum degree ; with high
probability Maker wins the Hamiltonicity game exactly at the time the random
graph process first reaches minimum degree . The latter two statements
settle conjectures of Stojakovi\'{c} and Szab\'{o}.Comment: 24 page
Local resilience and Hamiltonicity Maker-Breaker games in random-regular graphs
For an increasing monotone graph property \mP the \emph{local resilience}
of a graph with respect to \mP is the minimal for which there exists
of a subgraph with all degrees at most such that the removal
of the edges of from creates a graph that does not possesses \mP.
This notion, which was implicitly studied for some ad-hoc properties, was
recently treated in a more systematic way in a paper by Sudakov and Vu. Most
research conducted with respect to this distance notion focused on the Binomial
random graph model \GNP and some families of pseudo-random graphs with
respect to several graph properties such as containing a perfect matching and
being Hamiltonian, to name a few. In this paper we continue to explore the
local resilience notion, but turn our attention to random and pseudo-random
\emph{regular} graphs of constant degree. We investigate the local resilience
of the typical random -regular graph with respect to edge and vertex
connectivity, containing a perfect matching, and being Hamiltonian. In
particular we prove that for every positive and large enough values
of with high probability the local resilience of the random -regular
graph, \GND, with respect to being Hamiltonian is at least .
We also prove that for the Binomial random graph model \GNP, for every
positive and large enough values of , if
then with high probability the local resilience of \GNP with respect to being
Hamiltonian is at least . Finally, we apply similar
techniques to Positional Games and prove that if is large enough then with
high probability a typical random -regular graph is such that in the
unbiased Maker-Breaker game played on the edges of , Maker has a winning
strategy to create a Hamilton cycle.Comment: 34 pages. 1 figur
Positional Games
Positional games are a branch of combinatorics, researching a variety of
two-player games, ranging from popular recreational games such as Tic-Tac-Toe
and Hex, to purely abstract games played on graphs and hypergraphs. It is
closely connected to many other combinatorial disciplines such as Ramsey
theory, extremal graph and set theory, probabilistic combinatorics, and to
computer science. We survey the basic notions of the field, its approaches and
tools, as well as numerous recent advances, standing open problems and
promising research directions.Comment: Submitted to Proceedings of the ICM 201
On the optimality of the uniform random strategy
The concept of biased Maker-Breaker games, introduced by Chv\'atal and Erd{\H
o}s, is a central topic in the field of positional games, with deep connections
to the theory of random structures. For any given hypergraph the
main questions is to determine the smallest bias that allows
Breaker to force that Maker ends up with an independent set of . Here
we prove matching general winning criteria for Maker and Breaker when the game
hypergraph satisfies a couple of natural `container-type' regularity conditions
about the degree of subsets of its vertices. This will enable us to derive a
hypergraph generalization of the -building games, studied for graphs by
Bednarska and {\L}uczak. Furthermore, we investigate the biased version of
generalizations of the van der Waerden games introduced by Beck. We refer to
these generalizations as Rado games and determine their threshold bias up to
constant factors by applying our general criteria. We find it quite remarkable
that a purely game theoretic deterministic approach provides the right order of
magnitude for such a wide variety of hypergraphs, when the generalizations to
hypergraphs in the analogous setup of sparse random discrete structures are
usually quite challenging.Comment: 26 page
Generating random graphs in biased Maker-Breaker games
We present a general approach connecting biased Maker-Breaker games and
problems about local resilience in random graphs. We utilize this approach to
prove new results and also to derive some known results about biased
Maker-Breaker games. In particular, we show that for
, Maker can build a pancyclic graph (that is, a graph
that contains cycles of every possible length) while playing a game on
. As another application, we show that for , playing a game on , Maker can build a graph which
contains copies of all spanning trees having maximum degree with
a bare path of linear length (a bare path in a tree is a path with all
interior vertices of degree exactly two in )
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