#### DMCA

## Lower bounds for the cop number when the robber is fast

Venue: | Combinatorics, Probability and Computing |

Citations: | 4 - 2 self |

### Citations

69 |
Vertex-to-vertex pursuit in a graph,
- NOWAKOWSKI, WINKLER
- 1983
(Show Context)
Citation Context ...lso conjecture a general upper bound O(nt/t+1) for the cop number in this variant, generalizing Meyniel’s conjecture. 1 Introduction The game of Cops and Robbers, introduced by Nowakowski and Winkler =-=[10]-=- and independently by Quilliot [11], is a perfect information game played on a finite graph G. There are two players, a set of cops and a robber. Initially, the cops are placed onto vertices of their ... |

52 |
Cops and robbers in graphs with large girth and Cayley graphs,
- Frankl
- 1987
(Show Context)
Citation Context ...nt. For a survey of results on the cop number and related search parameters, see the survey by Hahn [7]. The most well known open question in this area is Meyniel’s conjecture, published by Frankl in =-=[5]-=-. It states that for every graph G on n vertices, O( √ n) cops are enough to win. This is asymptotically tight, i.e. for every n there exists an n-vertex graph with cop number Ω( √ n). The best upper ... |

29 |
Jeux et pointes fixes sur les graphes
- Quilliot
- 1978
(Show Context)
Citation Context ...d O(nt/t+1) for the cop number in this variant, generalizing Meyniel’s conjecture. 1 Introduction The game of Cops and Robbers, introduced by Nowakowski and Winkler [10] and independently by Quilliot =-=[11]-=-, is a perfect information game played on a finite graph G. There are two players, a set of cops and a robber. Initially, the cops are placed onto vertices of their choice in G (where more than one co... |

16 |
Dynamic cage survey, Electron
- Exoo, Jajcay
(Show Context)
Citation Context ...r to use Corollary 1 to prove interesting lower bounds for the cop number, one should look at vertex-minimal regular graphs with large girth, known as cages. Here are two useful results on cages (see =-=[3]-=- for a survey): Theorem 1 ([8]). Let g ≥ 5, and d ≥ 3 be an odd prime power. Then there exists a d-regular graph of girth g with at most 2d1+ 3 4 g−a vertices, where a = 4, 11/4, 7/2, 13/4 for g ≡ 0, ... |

9 | On a generalization of meyniel’s conjecture on the cops and robbers game. Research report - Alon, Mehrabian - 2010 |

9 |
Pursuing a fast robber on a graph, Theoret
- Fomin, Golovach, et al.
(Show Context)
Citation Context ... she is not allowed to pass through a vertex occupied by a cop. The parameter t is called the speed of the robber. This variant was first considered by Fomin, Golovach, Kratochv́ıl, Nisse, and Suchan =-=[4]-=-, who proved that computing the cop number is NP-hard for every t. Next, Frieze, Krivelevich, and Loh [6] showed that the cop number of an n-vertex graph can be as large as Ω(n t−3 t−2 ). They also as... |

7 |
New upper bounds on the order of cages, Electron
- Lazebnik, Ustimenko, et al.
- 1997
(Show Context)
Citation Context ...interesting lower bounds for the cop number, one should look at vertex-minimal regular graphs with large girth, known as cages. Here are two useful results on cages (see [3] for a survey): Theorem 1 (=-=[8]-=-). Let g ≥ 5, and d ≥ 3 be an odd prime power. Then there exists a d-regular graph of girth g with at most 2d1+ 3 4 g−a vertices, where a = 4, 11/4, 7/2, 13/4 for g ≡ 0, 1, 2, 3 (mod 4), respectively.... |

6 |
On Meyniel’s conjecture of the cop number, submitted
- Lu, Peng
- 2009
(Show Context)
Citation Context ... O( √ n) cops are enough to win. This is asymptotically tight, i.e. for every n there exists an n-vertex graph with cop number Ω( √ n). The best upper bound found so far is n2−(1−o(1)) √ log 2 n (see =-=[6, 9, 12]-=- for several proofs). 1 Here we consider the variant where in each move, the robber can take any path of length at most t from her current position, but she is not allowed to pass through a vertex occ... |

5 |
A new bound for the cops and robbers problem, arXiv:1004.2010v1 [math.CO
- Scott, Sudakov
(Show Context)
Citation Context ... O( √ n) cops are enough to win. This is asymptotically tight, i.e. for every n there exists an n-vertex graph with cop number Ω( √ n). The best upper bound found so far is n2−(1−o(1)) √ log 2 n (see =-=[6, 9, 12]-=- for several proofs). 1 Here we consider the variant where in each move, the robber can take any path of length at most t from her current position, but she is not allowed to pass through a vertex occ... |

2 |
Serra,On upper bounds and connectivity of cages, Australas
- Araujo, González, et al.
(Show Context)
Citation Context ..., and d ≥ 3 be an odd prime power. Then there exists a d-regular graph of girth g with at most 2d1+ 3 4 g−a vertices, where a = 4, 11/4, 7/2, 13/4 for g ≡ 0, 1, 2, 3 (mod 4), respectively. Theorem 2 (=-=[2]-=-). Let d ≥ 3 be a prime power. Then there exists a d-regular graph with girth 12 and at most 2d5 vertices. Theorem 3. Let t be some fixed positive integer denoting the speed of the robber. (a) If t ≥ ... |