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A proof of the Meyniel conjecture for Abelian Cayley graphs

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Peter Bradshaw at Simon Fraser University
Peter Bradshaw
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Abstract

We prove that the cop number of a connected abelian Cayley graph on n vertices is bounded by 7n. This proves that H. Meyniel’s conjectured bound of O(n) for the cop number of any connected graph on n vertices holds for abelian Cayley graphs.

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... The concept of cop number was introduced shortly afterwards by Aigner and Fromme in [1]. The cop number is well-studied on many classes of graphs; bounds are known, for example, for graphs of high girth [7], Cayley graphs [6,7,8,12], intersection graphs [10], and graphs with certain forbidden subgraphs [17,22]. ...
... Meyniel's conjecture is known, for example, to hold for undirected graphs of diameter 2 [21,28]. The first author has also shown in [6] that the cop number of Cayley graphs on abelian groups satisfies Meyniel's conjecture, with an upper bound of 7 √ n. Of course, the cop number is bounded above by a constant for many graph classes, such as graphs of bounded genus [1,26,27], graphs of bounded treewidth [17] and graphs without long induced paths [17]. ...
... Of course, the cop number is bounded above by a constant for many graph classes, such as graphs of bounded genus [1,26,27], graphs of bounded treewidth [17] and graphs without long induced paths [17]. In this paper, we will generalize the methods of [6] and [8] to both improve the upper bound for the cop number of Cayley graphs on abelian groups and show that directed Cayley graphs on abelian groups also satisfy Meyniel's conjecture, which will make these graph classes among the few large classes known to satisfy the conjecture. ...
Cops and robbers on directed and undirected abelian Cayley graphs
Preprint
Full-text available
  • Sep 2019
We show that the cop number of directed and undirected Cayley graphs on abelian groups has an upper bound of the form of O(n)O(\sqrt{n}), where n is the number of vertices, by introducing a refined inductive method. With our method, we improve the previous upper bound on cop number for undirected Cayley graphs on abelian groups, and we establish an upper bound on the cop number of directed Cayley graphs on abelian groups. We also use Cayley graphs on abelian groups to construct new \emph{Meyniel extremal families}, which contain graphs of every order n with cop number Θ(n)\Theta(\sqrt{n}).
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... In recent decades cops and robbers has seen a remarkable amount of attention with many general results as well as results for specific graph classes appearing in the literature. For instance, the cop number of planar graphs [1], graphs of higher genus [6,8,27,28], Cayley graphs [9,15], graph products [22], and random graphs [4,21,25] have all been extensively studied. Significant attention has also been paid to computational questions involving cop number, with MacGillivray and Clarke [13] showing that deciding if a graph is k-cop win is fixed parameter polynomial time in the order of the graph and k, while Kinnersley [17] proved that determining if c(G) ≤ k is EXPTIME-complete when k is not fixed. ...
... If correct, then this bound could not be improved as there are known graph families with c(G) = Ω( √ n) [5,24]. Despite Meyniel's conjecture being resolved for a number of graph classes such as Abelian Cayley graphs [9] and random graphs [4,25], the larger conjecture remains widely open despite significant effort [11,16,19,29]. In fact, it remains to be shown that there exists a α > 0 such that c(G) = O(n 1−α ), with this problem sometimes begin dubbed the weak Meyniel's conjecture. ...
Graphs with Large Girth and Small Cop Number
Preprint
Full-text available
  • May 2023
In this paper we consider the cop number of graphs with no, or few, short cycles. We show that when the girth of G is at least 8 and the minimum degree is sufficiently large, δ≥(ln⁡n)11−α\delta \geq (\ln{n})^{\frac{1}{1-\alpha}} where α∈(0,1)\alpha \in (0,1), then c(G)=o(nδβ−⌊g4⌋)c(G) = o(n \delta^{\beta -\lfloor \frac{g}{4} \rfloor}) as δ→∞\delta \rightarrow \infty where β>1−α\beta> 1-\alpha. This extends work of Frankl and implies that if G is large and dense in the sense that δ≥n2g−o(1)\delta \geq n^{\frac{2}{g} - o(1)} while also having girth g≥8g \geq 8, then G satisfies Meyniel's conjecture, that is c(G)=O(n)c(G) = O(\sqrt{n}). Moreover, it implies that if G is large and dense in the sense that there δ≥nϵ\delta \geq n^{\epsilon} for some ϵ>0\epsilon >0, while also having girth g≥8g \geq 8, then there exists an α>0\alpha>0 such that c(G)=O(n1−α)c(G) = O(n^{1-\alpha}), thereby satisfying the weak Meyniel's conjecture. Of course, this implies similar results for dense graphs with small, that is O(n1−α)O(n^{1-\alpha}), numbers of short cycles, as each cycle can be broken by adding a single cop. We also, show that there are graphs G with girth g and minimum degree δ\delta such that the cop number is at most o(g(δ−1)(1+o(1))g4)o(g (\delta-1)^{(1+o(1))\frac{g}{4}}). This resolves a recent conjecture by Bradshaw, Hosseini, Mohar, and Stacho, by showing that the constant 14\frac{1}{4} cannot be improved in the exponent of a lower bound c(G)≥1g(δ−1)⌊g−14⌋c(G) \geq \frac{1}{g} (\delta - 1)^{\lfloor \frac{g-1}{4}\rfloor}.
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... The best known upper bound, proved independently by [12,16,23], says that the cop number of any graph on n vertices is upper bounded by n 2 ð1þoð1ÞÞ ffiffiffiffiffiffi ffi logðnÞ p . Sharper results are known for special classes of graphs, such as random graphs [3,6,7,17,19], planar graphs [13], graphs with bounded genus [21,22], Cayley graphs [8,11], and more. For surveys of known related results see [1,5]. ...
... Frankl [11] proved that for any connected abelian Cayley graphs it holds that cðCðG; SÞÞ dðjSj þ 1Þ=2e. Recently, Bradshaw [8] showed that the cop number of any connected abelian Cayley graph on n vertices is bounded by 7 ffiffi ffi n p ...
Meyniel Extremal Families of Abelian Cayley Graphs
Article
Full-text available
  • Jun 2022
  • GRAPH COMBINATOR
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)O(\sqrt{n}). That is, for every graph G on n vertices O(n)O(\sqrt{n}) cops 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)O(\sqrt{n}). This proves that the O(n)O(\sqrt{n}) upper bound for Cayley graphs proved by Bradshaw (Discret Math 343:1, 2019) is tight. In particular, this shows that Meyniel’s conjecture, if true, is tight even for abelian Cayley graphs. In order to prove the result, we construct Cayley graphs on n vertices with Ω(n)\Omega (\sqrt{n}) generators that are K2,3K_{2,3}-free. This shows that the Kövári, Sós, and Turán theorem, stating that any K2,3K_{2,3}-free graph of n vertices has at most O(n3/2)O(n^{3/2}) edges, is tight up to a multiplicative constant even for abelian Cayley graphs.
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... The best known upper bound, proved independently by [14,22,11], says that the cop number of any graph on n vertices is upper bounded by n/2 (1+o(1)) √ 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]. ...
... 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 √ n. In this work we prove a lower bound that matches ...
Meyniel Extremal Families of Abelian Cayley Graphs
Preprint
  • Sep 2019
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)O(\sqrt{n}), i.e., that O(n)O(\sqrt{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)O(\sqrt{n}). This proves that the O(n)O(\sqrt{n}) upper bound for Cayley graphs proved by Bradshaw 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)\Omega(\sqrt{n}) generators that are K2,3K_{2,3}-free. This shows that the K\"{o}v\'{a}ri, S\'{o}s, and Tur\'{a}n theorem, stating that any K2,3K_{2,3}-free graph of n vertices has at most O(n3/2)O(n^{3/2}) edges, is tight up to a multiplicative constant even for abelian Cayley graphs.
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... The Cops and Robber game is a pursuit-evasion game. The game is commonly played on graphs [1,5,6,7,9,12,16,19], and as a new variant on geodesic spaces [15,21,22]. The players in the Cops and Robber game are the robber r and k cops c 1 , . . . ...
Cops and Robber on Hyperbolic Manifolds
Conference Paper
Full-text available
  • Jan 2023
The Cops and Robber game on geodesic spaces is a pursuit-evasion game with discrete steps which captures the behavior of the game played on graphs, as well as that of continuous pursuit-evasion games. One of the outstanding open problems about the game on graphs is to determine which graphs embeddable in a surface of genus g have largest cop number. It is known that the cop number of genus g graphs is O(g) and that there are examples whose cop number is Ω~(g )\tilde\Omega(\sqrt{g}\,). The same phenomenon occurs when the game is played on geodesic surfaces. In this paper we obtain a surprising result when the game is played on a surface with constant curvature. It is shown that two cops have a strategy to come arbitrarily close to the robber, independently of the genus. For special hyperbolic surfaces we also give upper bounds on the number of cops needed to catch the robber. Our results generalize to higher-dimensional hyperbolic manifolds.
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... More recent results on the cop number problem can be found in the works of Schroeder [38], Lu-Peng [32], Scott-Sudakov [39] etc. Recently, Bradshaw [12] and Bradshaw-Hosseini-Turcotte [14] proved the Meyniel's conjecture for abelian Cayley graphs. In fact, there has been a flurry of further activities on the determination of the cop number for classes of graphs. ...
On the cop number and the weak Meyniel's conjecture for algebraic graphs
Preprint
  • Mar 2023
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 Bollob\'{a}s--Janson--Riordan, we show that the weak Meyniel's conjecture holds for these algebraic graphs.
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Cops and robbers on directed and undirected abelian Cayley graphs
Article
  • Oct 2021
  • EUR J COMBIN
We show that the cop number of directed and undirected Cayley graphs on abelian groups is in O(n), where n is the number of vertices, by introducing a refined inductive method. With our method, we improve the previous upper bound on cop number for undirected Cayley graphs on abelian groups, and we establish an upper bound on the cop number of directed Cayley graphs on abelian groups. We also use Cayley graphs on abelian groups to construct new Meyniel extremal families, which contain graphs of every order n with cop number in Θ(n).
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A Bound for the Cops and Robbers Problem
Article
Full-text available
  • Apr 2010
In this short paper we study the game of cops and robbers, which is played on the vertices of some fixed graph G. Cops and a robber are allowed to move along the edges of G and the goal of cops is to capture the robber. The cop number c(G) of G is the minimum number of cops required to win the game. Meyniel conjectured a long time ago that O(n)O(\sqrt{n}) cops are enough for any connected G on n vertices. Improving several previous results, we prove that the cop number of n-vertex graph is at most n2−(1+o(1))log⁡nn 2^{-(1+o(1))\sqrt{\log n}}. Comment: 4 pages
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On Meyniel's conjecture of the cop number
Article
  • Oct 2012
  • J GRAPH THEOR
Meyniel conjectured that the cop number c(G) of any connected graph G on n vertices is at most for some constant C. In this article, we prove Meyniel's conjecture in special cases that G has diameter 2 or G is a bipartite graph of diameter 3. For general connected graphs, we prove , improving the best previously known upper-bound O(n/ lnn) due to Chiniforooshan. © 2012 Wiley Periodicals, Inc. (Contract grant sponsor: NSF; contract grant numbers: DMS 0701111; DMS 1000475 (to L. L. and X. P.).)
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On a pursuit game on cayley graphs
Article
  • Jan 1987
The game cops and robbers is considered on Cayley graphs of abelian groups. It is proved that if the graph has degreed, then [(d+1)/2] cops are sufficient to catch one robber. This bound is often best possible.
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Vertex-to-vertex Pursuit in a Graph
Article
  • Dec 1983
  • DISCRETE MATH
A graph G is given and two players, a cop and a robber, play the following game: the cop chooses a vertex, then the robber chooses a vertex, then the players move alternately beginning with the cop. A move consists of staying at one's present vertex or moving to an adjacent vertex; each move is seen by both players. The cop wins if he manages to occupy the same vertex as the robber, and the robber wins if he avoids this forever.We characterize the graphs on which the cop has a winning strategy, and connect the problem with the structure theory of graphs based on products and retracts.
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On a game of policemen and robber
Article
  • Jun 1987
  • DISCRETE APPL MATH
We consider a game where policemen try to catch a robber on a graph G (as defined by A. Quilliot) and we find the exact minimal number of policemen needed when G is a Cartesian product of trees.
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Cops and robbers in graphs with large girth and Cayley graphs
Article
  • Jun 1987
  • DISCRETE APPL MATH
It is shown that if a graph has girth at least 8t−3 and minimum degree greater that d, then more than dt cops are needed to catch a robber. Some upper bounds, in particular for Cayley graphs of groups, are also obtained.
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A Game of Cops and Robbers
Article
  • Apr 1984
  • DISCRETE APPL MATH
Let G be a finite connected graph. Two players, called cop C and robber R, play a game on G according to the following rules. First C then R occupy some vertex of G. After that they move alternately along edges of G. The cop C wins if he succeeds in putting himself on top of the robber R, otherwise R wins. We review an algorithmic characterization and structural description due to Nowakowski and Winkler. Then we consider the general situation where n cops chase the robber. It is shown that there are graphs on which arbitrarily many cops are needed to catch the robber. In contrast to this result, we prove that for planar graphs 3 cops always suffice to win.
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Problèmes de jeux, de point fixe, de connectivité et de représentation sur des graphes, des ensembles ordonnés et des hypergraphes
  • Jan 1978
  • A Quilliot
A. Quilliot, Problèmes de jeux, de point fixe, de connectivité et de représentation sur des graphes, des ensembles ordonnés et des hypergraphes (Ph.D. thesis), Université de Paris VI, 1978.