4 research outputs found
Chasing robbers on random geometric graphs---an alternative approach
We study the vertex pursuit game of \emph{Cops and Robbers}, in which cops
try to capture a robber on the vertices of the graph. The minimum number of
cops required to win on a given graph is called the cop number of . We
focus on , a random geometric graph in which vertices are
chosen uniformly at random and independently from , and two vertices
are adjacent if the Euclidean distance between them is at most . The main
result is that if then the cop number is
with probability that tends to as tends to infinity. The case was
proved earlier and independently in \cite{bdfm}, using a different approach.
Our method provides a tight upper bound for the number of rounds
needed to catch the robber.Comment: 6 page
Meyniel's conjecture holds for random graphs
In the game of cops and robber, the cops try to capture a robber moving on
the vertices of the graph. The minimum number of cops required to win on a
given graph is called the cop number of . The biggest open conjecture in
this area is the one of Meyniel, which asserts that for some absolute constant
, the cop number of every connected graph is at most .
In this paper, we show that Meyniel's conjecture holds asymptotically almost
surely for the binomial random graph. We do this by first showing that the
conjecture holds for a general class of graphs with some specific
expansion-type properties. This will also be used in a separate paper on random
-regular graphs, where we show that the conjecture holds asymptotically
almost surely when .Comment: revised versio
Acquaintance time of random graphs near connectivity threshold
Benjamini, Shinkar, and Tsur stated the following conjecture on the
acquaintance time: asymptotically almost surely for a random graph , provided that is connected. Recently,
Kinnersley, Mitsche, and the second author made a major step towards this
conjecture by showing that asymptotically almost surely , provided that has a Hamiltonian cycle. In this paper, we finish the
task by showing that the conjecture holds in the strongest possible sense, that
is, it holds right at the time the random graph process creates a connected
graph. Moreover, we generalize and investigate the problem for random
hypergraphs
First measurement of the cross section for top-quark pair production in proton-proton collisions at =7 TeV
The first measurement of the cross section for top-quark pair production in
pp collisions at the LHC at center-of-mass energy sqrt(s)= 7 TeV has been
performed using 3.1 {\pm} 0.3 inverse pb of data recorded by the CMS detector.
This result utilizes the final state with two isolated, highly energetic
charged leptons, large missing transverse energy, and two or more jets.
Backgrounds from Drell-Yan and non-W/Z boson production are estimated from
data. Eleven events are observed in the data with 2.1 {\pm} 1.0 events expected
from background. The measured cross section is 194 {\pm} 72 (stat.) {\pm} 24
(syst.) {\pm} 21 (lumi.) pb, consistent with next-to-leading order predictions