A better bound for the cop number of general graphs

Abstract

AbstractIn 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