Lower Bounds for the Cop Number When the Robber is Fast

Details Lower Bounds for the Cop Number When the Robber is Fast Lower Bounds for the Cop Number When the Robber is Fast

2010-07-10

arxiv.org/abs/1007.1734v2

Combinatorics, Probability and Computing (2011) 20, 617-621

Mathematics

Combinatorics

5 pages

Scientific paper

10.1017/S0963548311000101

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 Omega(d^t). 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 Omega(n^{2/3}) if t>1, and Omega(n^{4/5}) if t>3. This improves the Omega(n^{(t-3)/(t-2)}) lower bound of Frieze, Krivelevich, and Loh (Variations on Cops and Robbers, J. Graph Theory, 2011) when 1

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