Meyniel's conjecture on graphs of bounded degree

12/15/2019

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by   Seyyed Aliasghar Hosseini, et al.

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The game of Cops and Robbers is a well known pursuit-evasion game played on graphs. It has been proved <cit.> that cubic graphs can have arbitrarily large cop number c(G), but the known constructions show only that the set {c(G) | G cubic} is unbounded. In this paper we prove that there are arbitrarily large subcubic graphs G whose cop number is at least n^1/2-o(1) where n=|V(G)|. We also show that proving Meyniel's conjecture for graphs of bounded degree implies a weak Meyniel's conjecture for all graphs.