A proof of the Meyniel conjecture for Abelian Cayley graphs

Bradshaw, Peter

doi:10.1016/j.disc.2019.06.002

Cited by 4 publications

(6 citation statements)

References 7 publications

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“…The concept of cop number was introduced shortly afterward by M. Aigner and M. Fromme [1]. The cop number parameter is well-studied on many classes of graphs; bounds are known for graph of bounded genus [17], graphs of high girth [6], Cayley graphs [7] [5], geometric graphs [8], and graphs with certain forbidden subgraphs [14].…”

Section: Introductionmentioning

confidence: 99%

“…P. Frankl first mentions Meyniel's conjecture with regard to undirected graphs in [6], and W. Baird and A. Bonato ask whether Meyniel's conjecture holds for strongly connected digraphs in [3]. Meyniel's conjecture is known, for example, to hold for projective plane incidence graphs [3] as well as for undirected abelian Cayley graphs [5]. In this paper, we will show that directed abelian Cayley graphs also satisfy Meyniel's conjecture, which will make this class one of the few large graph classes known to satisfy the conjecture.…”

Section: Introductionmentioning

confidence: 99%

“…In order to estimate the number of elements of G needed to account for all of T , we may iteratively build a set K whose elements account for all of T and then estimate the size of K, as P. Bradshaw does in [5]. We will build K one element at a time.…”

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confidence: 99%

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Cops and robbers on directed and undirected abelian Cayley graphs

Bradshaw,

Hosseini,

Turcotte

2019

Preprint

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Cops and robbers on directed and undirected abelian Cayley graphs

Bradshaw,

Hosseini,

Turcotte

2019

Preprint

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“…√ n . Sharper results are known for special classes of graphs, such as random graphs [3,4,5,15,18], planar graphs [16], graphs with bounded genus [20,21], Cayley graphs [8,10], and more. For a survey of known related results see [7].…”

Section: Introductionmentioning

confidence: 99%

“…Frankl [10] proved that for any connected abelian Cayley graphs it holds that c(C(G, S)) ≤ ⌈(|S| + 1)/2⌉. Recently, Bradshaw [8] showed that the cop number of any connected abelian Cayley graph on n vertices is bounded by 7…”

Section: Introductionmentioning

confidence: 99%

Meyniel Extremal Families of Abelian Cayley Graphs

Hasiri¹,

Shinkar²

2019

Preprint

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We study the game of Cops and Robbers, where cops try to capture a robber on the vertices of a graph. Meyniel's conjecture states that for every connected graph G on n vertices, the cop number of G is upper bounded by O( √ n), i.e., that O( √ n) suffice to catch the robber. We present several families of abelian Cayley graphs that are Meyniel extremal, i.e., graphs whose cop number is O( √ n). This proves that the O( √ n) upper bound for Cayley graphs proved by Bradshaw [8] is tight up to a multiplicative constant. In particular, this shows that Meyniel's conjecture, if true, is tight to a multiplicative constant even for abelian Cayley graphs.In order to prove the result, we construct Cayley graphs on n vertices with Ω( √ n) generators that are K 2,3 -free. This shows that the Kövári, Sós, and Turán theorem, stating that any K 2,3free graph of n vertices has at most O(n 3/2 ) edges, is tight up to a multiplicative constant even for abelian Cayley graphs.

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On the cop number and the weak Meyniel's conjecture for algebraic graphs

Biswas¹,

Saha²

2023

Preprint

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In this article, we study the game of cops and robbers in algebraic graphs. We show that the cop number of the Cayley sum graph of a finite group G with respect to a subset S is at most its degree when the graph is connected, undirected. We also show that a similar bound holds for the cop number of generalised Cayley graphs and the twisted Cayley sum graphs under some conditions. These extend a result of Frankl to such graphs. Using the above bounds and a result of Bollobás-Janson-Riordan, we show that the weak Meyniel's conjecture holds for these algebraic graphs.

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A proof of the Meyniel conjecture for Abelian Cayley graphs

Cited by 4 publications

References 7 publications

Cops and robbers on directed and undirected abelian Cayley graphs

Cops and robbers on directed and undirected abelian Cayley graphs

Meyniel Extremal Families of Abelian Cayley Graphs

On the cop number and the weak Meyniel's conjecture for algebraic graphs

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