Meyniel’s conjecture on the cop number: A survey

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.

[1]  Tchébichef,et al.  Mémoire sur les nombres premiers. , 1852 .

[2]  Paul Erdös,et al.  On random graphs, I , 1959 .

[3]  J. Moon.  Topics on tournaments , 1968 .

[4]  Peter Winkler,et al.  Vertex-to-vertex pursuit in a graph , 1983, Discret. Math..

[5]  B. Bollobás.  The evolution of random graphs , 1984 .

[6]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[7]  Martin Aigner,et al.  A game of cops and robbers , 1984, Discret. Appl. Math..

[8]  Thomas ANDREAE,et al.  Note on a pursuit game played on graphs , 1984, Discret. Appl. Math..

[9]  Alain Quilliot,et al.  A short note about pursuit games played on a graph with a given genus , 1985, J. Comb. Theory, Ser. B.

[10]  Thomas Andreae,et al.  On a pursuit game played on graphs for which a minor is excluded , 1986, J. Comb. Theory, Ser. B.

[11]  Peter Frankl,et al.  Cops and robbers in graphs with large girth and Cayley graphs , 1987, Discret. Appl. Math..

[12]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[13]  D. West.  Introduction to Graph Theory , 1995 .

[14]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[15]  A. Rbnyi.  ON THE EVOLUTION OF RANDOM GRAPHS , 2001 .

[16]  Bernd S. W. Schröder.  The Copnumber of a Graph is Bounded by [3/2 genus ( G )] + 3 , 2001 .

[17]  Vadim Bulitko,et al.  A Cover-Based Approach to Multi-Agent Moving Target Pursuit , 2008, AIIDE.

[18]  E. Chiniforooshan.  A better bound for the cop number of general graphs , 2008 .

[19]  Nathan R. Sturtevant,et al.  Evaluating Strategies for Running from the Cops , 2009, IJCAI.

[20]  A. Scott,et al.  A new bound for the cops and robbers problem , 2010, 1004.2010.

[21]  Anthony Bonato,et al.  Cops and Robbers from a distance , 2010, Theor. Comput. Sci..

[22]  Pawel Pralat.  When does a random graph have constant cop number? , 2010, Australas. J Comb..

[23]  Benny Sudakov,et al.  A Bound for the Cops and Robbers Problem , 2011, SIAM J. Discret. Math..

[24]  Anthony Bonato,et al.  The Game of Cops and Robbers on Graphs , 2011 .

[25]  Alan M. Frieze,et al.  Variations on cops and robbers , 2010, J. Graph Theory.

[26]  Anthony Bonato,et al.  Almost all cop-win graphs contain a universal vertex , 2012, Discret. Math..

[27]  Linyuan Lu,et al.  On Meyniel's conjecture of the cop number , 2012, J. Graph Theory.

[28]  Anthony Bonato,et al.  Cops and Robbers on Graphs Based on Designs , 2013 .

[29]  Béla Bollobás,et al.  Cops and robbers in a random graph , 2013, J. Comb. Theory, Ser. B.