Lower Bounds for the Cop Number When the Robber is Fast

TL;DR

This paper investigates a variant of the Cops and Robbers game where the robber moves multiple edges at once, establishing new lower bounds on the cop number for certain graph classes and proposing a generalized conjecture.

Contribution

It introduces a lower bound for the cop number in the fast robber variant and conjectures a general upper bound extending Meyniel's conjecture.

Findings

Cop number is Omega(d^t) for certain d-regular graphs with large girth.

Cop number can be as large as Omega(n^{2/3}) or Omega(n^{4/5}) depending on t.

Proposes a conjecture for an upper bound O(n^{t/(t+1)}) for the cop number.

Abstract

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<t<7. We also conjecture a general upper bound O(n^{t/t+1}) for the cop number in this variant, generalizing Meyniel's conjecture.