On the cop number of generalized Petersen graphs

Ball, Taylor; Bell, Robert W.; Guzman, Jonathan; Hanson-Colvin, Madeleine; Schonsheck, Nikolas

doi:10.1016/j.disc.2016.10.004

Cited by 7 publications

(30 citation statements)

References 9 publications

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“…We intend to demonstrate that three cops do not have a winning strategy on the graphs we are studying. Together with the upper bound of [1], this is sufficient to show that the cop number for these graphs is 4. In order to prove that three cops do not have a winning strategy, we will first show that if three cops have not already trapped the robber, the robber always has a legal move whereby the cops cannot trap the robber on the cops' next move.…”

Section: Resultsmentioning

confidence: 91%

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On generalised Petersen graphs of girth 7 that have cop number 4

Morris

2022

Art Discrete Appl. Math.

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We show that if n = 7k/i with i ∈ {1, 2, 3} then the cop number of the generalised Petersen graph GP (n, k) is 4, with some small previously-known exceptions. It was previously proved by Ball et al. (2015) that the cop number of any generalised Petersen graph is at most 4. The results in this paper explain all of the known generalised Petersen graphs that actually have cop number 4 but were not previously explained by Morris et al. in a recent preprint, and places them in the context of infinite families. (More precisely, the preprint by Morris et al. explains all known generalised Petersen graphs with cop number 4 and girth 8, while this paper explains those that have girth 7.

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Section: Resultsmentioning

confidence: 91%

“…In [1], Ball et al showed that the cop number of every generalised Petersen graph is at most 4. They also provided a list of all generalised Petersen graphs with n ≤ 40 that attain this bound.…”

Section: Introductionmentioning

confidence: 99%

On generalised Petersen graphs of girth 7 that have cop number 4

Morris

2022

Art Discrete Appl. Math.

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“…We focus on generalized Peterson graphs in this paper, although our main result applies to any cubic graph of girth 8. In [1] the analysis was extended to I graphs. They proved that the cop number of a connected I-graph I(n, k, j) is less than or equal to 5.…”

Section: Previous Researchmentioning

confidence: 99%

“…For some families that do not have girth 8, the cop number is already known, or tighter bounds have been found. For example, when k = 1, GP (n, 1) has a girth of 4, and its cop number is known to be 2 [1]. Likewise, when k = 3 the girth is 6 (except for some small values of n) and the cop number is known to be at most 3 [1].…”

Section: Previous Researchmentioning

confidence: 99%

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Most Generalized Petersen graphs of girth 8 have cop number 4

Morris¹,

Runte²,

Adrian³

2020

Preprint

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A generalized Petersen graph GP (n, k) is a regular cubic graph on 2n vertices (the parameter k is used to define some of the edges). It was previously shown (Ball et al., 2015) that the cop number of GP (n, k) is at most 4, for all permissible values of n and k. In this paper we prove that the cop number of "most" generalized Petersen graphs is exactly 4. More precisely, we show that unless n and k fall into certain specified categories, then the cop number of GP (n, k) is 4. The graphs to which our result applies all have girth 8.In fact, our argument is slightly more general: we show that in any cubic graph of girth at least 8, unless there exist two cycles of length 8 whose intersection is a path of length 2, then the cop number of the graph is at least 4. Even more generally, in a graph of girth at least 9 and minimum valency δ, the cop number is at least δ + 1.

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