A better bound for the cop number of general graphs

  • Authors:

  • Ehsan Chiniforooshan

  • Affiliations:

  • School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2008

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Abstract

In this note, we prove that the cop number of any n-vertex graph G, denoted by ${{c}}({{G}})$, is at most ${{O}}\big({{{n}}\over {{\rm lg}} {{n}}}\big)$. Meyniel conjectured ${{c}}({{G}})={{O}}(\sqrt{{{n}}})$. It appears that the best previously known sublinear upper-bound is due to Frankl, who proved ${{c}}({{G}})\leq ({{1}}+ {{o}}({{1}})){{{n}}{{\rm lg}}{{\rm lg}} {{n}}\over {{\rm lg}} {{n}}}$. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 45–48, 2008