8 research outputs found
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
On the minimum order of -cop win graphs
We consider the minimum order graphs with a given cop number. We prove that the minimum order of a connected graph with cop number 3 is 10, and show that the Petersen graph is the unique isomorphism type of graph with this property. We provide the results of a computational search on the cop number of all graphs up to and including order 10. A relationship is presented between the minimum order of graph with cop number and Meyniel's conjecture on the asymptotic maximum value of the cop number of a connected graph
Lower Bounds for the Cop Number When the Robber is Fast
We consider a variant of the Cops and Robbers game where the robber can move
t edges at a time, and show that in this variant, the cop number of a d-regular
graph with girth larger than 2t+2 is Omega(d^t). By the known upper bounds on
the order of cages, this implies that the cop number of a connected n-vertex
graph can be as large as Omega(n^{2/3}) if t>1, and Omega(n^{4/5}) if t>3. This
improves the Omega(n^{(t-3)/(t-2)}) lower bound of Frieze, Krivelevich, and Loh
(Variations on Cops and Robbers, J. Graph Theory, 2011) when 1<t<7. We also
conjecture a general upper bound O(n^{t/t+1}) for the cop number in this
variant, generalizing Meyniel's conjecture.Comment: 5 page
Cops and Robber Game with a Fast Robber
Graph searching problems are described as games played on graphs, between a set of searchers and a fugitive. Variants of the game restrict the abilities of the searchers and the fugitive and the corresponding search number (the least number of searchers that have a winning strategy) is related to several well-known parameters in graph theory. One popular variant is called the Cops and Robber game, where the searchers (cops) and the fugitive (robber) move in rounds, and in each round they move to an adjacent vertex. This game, defined in late 1970's, has been studied intensively. The most famous open problem is Meyniel's conjecture, which states that the cop number
(the minimum number of cops that can always capture the robber) of a connected graph
on n vertices is O(sqrt n).
We consider a version of the Cops and Robber game, where the robber is faster than the cops, but is not allowed to jump over the cops. This version was first studied in 2008.
We show that when the robber has speed s,
the cop number of a connected n-vertex graph can be as large as Omega(n^(s/s+1)). This improves the Omega(n^(s-3/s-2)) lower bound of Frieze, Krivelevich, and Loh (Variations on Cops and Robbers, J. Graph Theory, to appear). We also conjecture a general upper bound O(n^(s/s+1)) for the cop number,
generalizing Meyniel's conjecture.
Then we focus on the version where the robber is infinitely fast, but is again not allowed to jump over the cops. We give a mathematical characterization for graphs with cop number one. For a graph with treewidth tw and maximum degree Delta,
we prove the cop number is between (tw+1)/(Delta+1) and tw+1. Using this we show that the cop number of the m-dimensional hypercube is
between c1 n / m sqrt(m) and c2 n / m for some constants c1 and c2. If G is a connected interval graph on n vertices, then we give a polynomial time 3-approximation algorithm for finding the cop number of G, and prove that the cop number is O(sqrt(n)).
We prove that given n, there exists a connected chordal graph on n vertices
with cop number Omega(n/log n). We show a lower bound for the cop numbers of expander graphs, and use this to prove that the random G(n,p) that is not very sparse,
asymptotically almost surely has cop number between d1 / p and d2 log (np) / p for suitable constants d1 and d2. Moreover, we prove that a fixed-degree regular random graph with n vertices asymptotically almost surely has cop number Theta(n)
On the minimum order of -cop win graphs
We consider the minimum order graphs with a given cop number. We prove that the minimum order of a connected graph with cop number 3 is 10, and show that the Petersen graph is the unique isomorphism type of graph with this property. We provide the results of a computational search on the cop number of all graphs up to and including order 10. A relationship is presented between the minimum order of graph with cop number and Meyniel's conjecture on the asymptotic maximum value of the cop number of a connected graph
Catching a robber on a random -uniform hypergraph
The game of \emph{Cops and Robber} is usually played on a graph, where a
group of cops attempt to catch a robber moving along the edges of the graph.
The \emph{cop number} of a graph is the minimum number of cops required to win
the game. An important conjecture in this area, due to Meyniel, states that the
cop number of an -vertex connected graph is . In 2016,
Pra{\l}at and Wormald [Meyniel's conjecture holds for random graphs, Random
Structures Algorithms. 48 (2016), no. 2, 396-421. MR3449604] showed that this
conjecture holds with high probability for random graphs above the
connectedness threshold. Moreoever, {\L}uczak and Pra{\l}at [Chasing robbers on
random graphs: Zigzag theorem, Random Structures Algorithms. 37 (2010), no. 4,
516-524. MR2760362] showed that on a -scale the cop number demonstrates a
surprising \emph{zigzag} behaviour in dense regimes of the binomial random
graph . In this paper, we consider the game of Cops and Robber on a
hypergraph, where the players move along hyperedges instead of edges. We show
that with high probability the cop number of the -uniform binomial random
hypergraph is for a
broad range of parameters and and that on a -scale our upper
bound on the cop number arises as the minimum of \emph{two} complementary
zigzag curves, as opposed to the case of . Furthermore, we conjecture
that the cop number of a connected -uniform hypergraph on vertices is
.Comment: 21 page