On Meyniel's conjecture of the cop number

Meyniel conjectured that the cop number c(G) of any connected graph G on n vertices is at most for some constant C. In this article, we prove Meyniel's conjecture in special cases that G has diameter 2 or G is a bipartite graph of diameter 3. For general connected graphs, we prove , improving the best previously known upper-bound O(n/ lnn) due to Chiniforooshan. © 2012 Wiley Periodicals, Inc. (Contract grant sponsor: NSF; contract grant numbers: DMS 0701111; DMS 1000475 (to L. L. and X. P.).)

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