37 research outputs found
Efficient winning strategies in random-turn Maker-Breaker games
We consider random-turn positional games, introduced by Peres, Schramm,
Sheffield and Wilson in 2007. A -random-turn positional game is a two-player
game, played the same as an ordinary positional game, except that instead of
alternating turns, a coin is being tossed before each turn to decide the
identity of the next player to move (the probability of Player I to move is
). We analyze the random-turn version of several classical Maker-Breaker
games such as the game Box (introduced by Chv\'atal and Erd\H os in 1987), the
Hamilton cycle game and the -vertex-connectivity game (both played on the
edge set of ). For each of these games we provide each of the players with
a (randomized) efficient strategy which typically ensures his win in the
asymptotic order of the minimum value of for which he typically wins the
game, assuming optimal strategies of both players.Comment: 20 page
A Note on the Minimum Number of Edges in Hypergraphs with Property O
An oriented -uniform hypergraph is said to have Property O if for every
linear order of the vertex set, there is some edge oriented consistently with
the linear order. Recently Duffus, Kay and R\"{o}dl investigated the minimum
number of edges in a -uniform hypergaph with Property O. They proved
that , where the upper bound holds for
sufficiently large. In this short note we improve their upper bound by a factor
of , showing that for every . We also
show that their lower bound is not tight. Furthermore, Duffus, Kay and R\"{o}dl
also studied the minimum number of vertices in a -uniform hypergaph
with Property O. For they showed , and asked for
the precise value of . Here we show .Comment: 6 pages, 1 figur
Goldberg's Conjecture is true for random multigraphs
In the 70s, Goldberg, and independently Seymour, conjectured that for any
multigraph , the chromatic index satisfies , where . We show that their conjecture (in a
stronger form) is true for random multigraphs. Let be the probability
space consisting of all loopless multigraphs with vertices and edges,
in which pairs from are chosen independently at random with
repetitions. Our result states that, for a given ,
typically satisfies . In
particular, we show that if is even and , then
for a typical . Furthermore, for a fixed
, if is odd, then a typical has
for , and
for .Comment: 26 page
Packing, counting and covering Hamilton cycles in random directed graphs
A Hamilton cycle in a digraph is a cycle that passes through all the vertices, where all the arcs are oriented in the same direction. The problem of finding Hamilton cycles in directed graphs is well studied and is known to be hard. One of the main reasons for this is that there is no general tool for finding Hamilton cycles in directed graphs comparable to the so-called Posá ‘rotation-extension’ technique for the undirected analogue. Let D(n, p) denote the random digraph on vertex set [n], obtained by adding each directed edge independently with probability p. Here we present a general and a very simple method, using known results, to attack problems of packing and counting Hamilton cycles in random directed graphs, for every edge-probability p > logC(n)/n. Our results are asymptotically optimal with respect to all parameters and apply equally well to the undirected case
Shotgun assembly of random graphs
In the graph shotgun assembly problem, we are given the balls of radius
around each vertex of a graph and asked to reconstruct the graph. We study the
shotgun assembly of the Erd\H{o}s-R\'enyi random graph from a
wide range of values of . We determine the threshold for reconstructibility
for each , extending and improving substantially on results of Mossel
and Ross for . For , we give upper and lower bounds that improve on
results of Gaudio and Mossel by polynomial factors. We also give a sharpening
of a result of Huang and Tikhomirov for .Comment: 36 pages, 3 figure