# Meyniel’s conjecture on the cop number: A survey

@article{Baird2012MeynielsCO,
title={Meyniel’s conjecture on the cop number: A survey},
author={William Baird and Anthony Bonato},
journal={The Journal of Combinatorics},
year={2012},
volume={3},
pages={225-238}
}
• Published 2012
• Mathematics
• The Journal of Combinatorics
Meyniel’s conjecture is one of the deepest open problems on the cop number of a graph. It states that for a connected graph G of order n, c(G) = O( √ n). While largely ignored for over 20 years, the conjecture is receiving increasing attention. We survey the origins of and recent developments towards the solution of the conjecture. We present some new results on Meyniel extremal families containing graphs of order n satisfying c(G) ≥ d√n, where d is a constant.

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#### References

SHOWING 1-10 OF 50 REFERENCES
On Meyniel's conjecture of the cop number
• Mathematics, Computer Science
• J. Graph Theory
• 2012
This article proves Meyniel's conjecture in special cases that G has diameter 2 or G is a bipartite graph of diameter 3, improving the best previously known upper-bound O(n/ lnn) due to Chiniforooshan. Expand
Cops and Robbers on Graphs Based on Designs
• Mathematics
• 2013
We investigate the cop number of graphs based on combinatorial designs. Incidence graphs, point graphs, and block intersection graphs are studied, with an emphasis on finding families of graphs withExpand
Note on a pursuit game played on graphs
A negative answer to the question if it is true that c(G) ≤ k whenever the maximal degree of G is at most k is given by showing that, for all positive integers k, n (k ≥ 3), there exists a k-regular graph G with c( G) ≥ n. Expand
A Bound for the Cops and Robbers Problem
• Computer Science, Mathematics
• SIAM J. Discret. Math.
• 2011
Improving several previous results, it is proved that the cop number of an n-vertex graph is at most n2 −ð1 þoð1ÞÞ ffiffiffiffIFFiffiffiffsiffiffi log n p . Expand
When does a random graph have constant cop number?
Asymptotic results for the game of Cops and Robbers played on a random graph G(n, p) focusing on the case when the cop number does not grow with the size of a graph are presented. Expand
Variations on cops and robbers
• Mathematics, Computer Science
• J. Graph Theory
• 2012
The directed graph version of the classical Cops and Robbers game is studied, and it is shown that the cop number of any strongly connected digraph on n vertices is O(n(loglogn)2/logn). Expand
On a pursuit game played on graphs for which a minor is excluded
• Thomas Andreae
• Computer Science, Mathematics
• J. Comb. Theory, Ser. B
• 1986
An upper bound for c(G) is established in terms of the cross-cap number of G, thus providing a (partial) analogue to a result of Quilliot on the genus. Expand
Cops and robbers in a random graph
• Computer Science, Mathematics
• J. Comb. Theory, Ser. B
• 2013
It is proved that for sparse random graphs the cop-number has order of magnitude n^1/^2^+^o^(^1^) cops, and it is shown that, for general graphs, this strategy cannot be too effective: there are graphs that need at least n^2/3/4/5 cops for this strategy. Expand
A short note about pursuit games played on a graph with a given genus
• A. Quilliot
• Computer Science, Mathematics
• J. Comb. Theory, Ser. B
• 1985
This result is extended, showing that if G is a graph with a given genus k, then 3 + 2k pursuers are enough to “arrest” the evader B. Expand
Cops and robbers in graphs with large girth and Cayley graphs
• P. Frankl
• Computer Science, Mathematics
• Discret. Appl. Math.
• 1987
Abstract It is shown that if a graph has girth at least 8 t −3 and minimum degree greater that d , then more than d t cops are needed to catch a robber. Some upper bounds, in particular for CayleyExpand