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Lower Bounds for the Cop Number when the Robber is Fast


We consider a variant of the Cops and Robbers game where the robber can move t edges at a time, and show that in this variant, the cop number of a d-regular graph with girth larger than 2t+2 is Ω(dt). By the known upper bounds on the order of cages, this implies that the cop number of a connected n-vertex graph can be as large as Ω(n2/3) if t ≥ 2, and Ω(n4/5) if t ≥ 4. This improves the Ω() lower bound of Frieze, Krivelevich and Loh (Variations on cops and robbers, J. Graph Theory, to appear) when 2 ≤ t ≤ 6. We also conjecture a general upper bound O(nt/t+1) for the cop number in this variant, generalizing Meyniel's conjecture.

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[4]F. V. Fomin , P. A. Golovach , J. Kratochvíl , N. Nisse and K. Suchan (2010) Pursuing a fast robber on a graph. Theoret. Comput. Sci. 411 11671181.

[5]P. Frankl (1987) Cops and robbers in graphs with large girth and Cayley graphs. Discrete Appl. Math. 17 301305.

[10]R. Nowakowski and P. Winkler (1983) Vertex-to-vertex pursuit in a graph. Discrete Math. 43 235239.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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