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On the cop number of graphs of high girth
Abstract
We establish a lower bound for the cop number of graphs of high girth in terms of the minimum degree, and more generally, in terms of a certain growth condition. We show, in particular, that the cop number of any graph with girth g g and minimum degree δ is at least 1 g ( δ − 1 ) ⌊ g − 1 4 ⌋ . We establish similar results for directed graphs. While exposing several reasons for conjecturing that the exponent 1 4 g in this lower bound cannot be improved to ( 1 4 + ε ) g , we are also able to prove that it cannot be increased beyond 3 8 g . This is established by considering a certain family of Ramanujan graphs. In our proof of this bound, we also show that the “weak” Meyniel's conjecture holds for expander graph families of bounded degree.
... Over the last 40 years Cops and Robbers has been extensively studied. In particular, the cop number of planar graphs [2] and graphs of large girth [9,12,16] have been studied and provide motivation for our research directions. There are a number of variants of the Cops and Robbers game within the literature, some of which affect the power dynamics between the cop player and the robber player. ...
... Fortunately, the graphs of smallest order with minimum degree 3 and girth 9 are known. These graphs are (3,9)-cages, the first of which was discovered in [4], with all 18 being later characterised in [10]. We adopt the labeling G 1 , . . . ...
... We adopt the labeling G 1 , . . . , G 18 of the (3,9)-cages given in [10]. All (3,9)-cages are order 58, and for each 1 ≤ i ≤ 18 we let H i = G 2 i − E(G i ). ...
This paper considers the Cops and Attacking Robbers game, a variant of Cops and Robbers, where the robber is empowered to attack a cop in the same way a cop can capture the robber. In a graph G, the number of cops required to capture a robber in the Cops and Attacking Robbers game is denoted by \attCop(G). We characterise the triangle-free graphs G with \attCop(G) \leq 2 via a natural generalisation of the cop-win characterisation by Nowakowski and Winkler \cite{nowakowski1983vertex}. We also prove that all bipartite planar graphs G have \attCop(G) \leq 4 and show this is tight by constructing a bipartite planar graph G with \attCop(G) = 4. Finally we construct 17 non-isomorphic graphs H of order 58 with \attCop(H) = 6 and \cop(H)=3. This provides the first example of a graph H with \attCop(H) - \cop(H) \geq 3 extending work by Bonato, Finbow, Gordinowicz, Haidar, Kinnersley, Mitsche, Pra\l{}at, and Stacho \cite{bonato2014robber}. We conclude with a list of conjectures and open problems.
... This lower bound was recently improved by Bradshaw, Hosseini, Mohar, and Stacho in [10] who showed that c(G) ≥ 1 g (δ − 1) ⌊ g−1 4 ⌋ and conjecture that the exponential coefficient of 1 4 cannot be improved. We prove in Theorem 1.4 that this conjecture is true. ...
... As the title suggests in this section we will prove Theorem 1.1. Our proof is in part probabilistic and proceeds similarly to arguments that appear in [10,19,29]. For readers unfamiliar with the probabilistic method we recommend [2] as a reference. ...
... In this section we will show that bipartite Ramanujan graphs introduced by Lubotsky, Phillips, and Sarnak in [20] have cop number at most (ε + o(1))gδ (1+o(1)) g 4 for every ε > 0. Note that these are the same graphs considered in [10] to show that there are graphs who's cop number is bounded above by (δ − 1) (1+o(1)) 3g 8 . Our argument employs a similar ideas to that of [10], hence we will use a number of the same lemmas. ...
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, where , then as where . This extends work of Frankl and implies that if G is large and dense in the sense that while also having girth , then G satisfies Meyniel's conjecture, that is . Moreover, it implies that if G is large and dense in the sense that there for some , while also having girth , then there exists an such that , thereby satisfying the weak Meyniel's conjecture. Of course, this implies similar results for dense graphs with small, that is , 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 such that the cop number is at most . This resolves a recent conjecture by Bradshaw, Hosseini, Mohar, and Stacho, by showing that the constant cannot be improved in the exponent of a lower bound .
... In fact, there has been a flurry of further activities on the determination of the cop number for classes of graphs. See the works of Lehner [30], Gonzalez Hermosillo de la Maza-Hosseini-Mohar [28], González Hermosillo de la Maza-Hosseini-Knox-Mohar-Reed [21], Das-Gahlawat-Sahoo-Sen [16], Bradshaw-Hosseini-Mohar-Stacho [13] among others. The bound given by Meyniel's conjecture is also asymptotically best possible, see Bradshaw-Hosseini-Turcotte [14,Section 5], Hasiri-Shinkar [27,Theorem 1] etc. ...
... The weak Meyniel's conjecture states that for a fixed constant c > 0, the cop number of a graph on n vertices is O(n 1−c ) [5, p. 227], [13, p. 33]. Bradshaw-Hosseini-Mohar-Stacho showed that the weak Meyniel's conjecture holds for expander families of bounded degree [13,Corollary 4.7]. In this section, we show that the weak Meyniel's conjecture holds for the Cayley graphs, the Cayley sum graphs, and the twisted Cayley and Cayley sum graphs under suitable hypotheses. ...
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.
... Also, for a class of graphs H, we say that G is H-free if G is H-free for every H ∈ H (the same definition can be stated for H-subgraph-free and H-minor-free). The first results in the context date back to 1984 when it was proved in [AF84] that all planar graphs have the cop number at most 3 and in [And84] that the class of k-regular graphs are cop-unbounded for every integer k ≥ 3. The latter result can be also implied from a recent result in [BHMS20] stating that the cop number of any graph with girth g and minimum degree δ is at least 1 g (δ − 1) g−1 4 ...
In this paper, we study the game of cops and robber on the class of graphs with no even hole (induced cycle of even length) and claw (a star with three leaves). The cop number of a graph G is defined as the minimum number of cops needed to capture the robber. Here, we prove that the cop number of all claw-free even-hole-free graphs is at most two and, in addition, the capture time is at most 2n rounds, where n is the number of vertices of the graph. Moreover, our results can be viewed as a first step towards studying the structure of claw-free even-hole-free graphs.
... Aigner and Fromme proved in [2] that graphs with girth at least 5 have a cop number of a least their minimum degree. This result has since been generalized by Frankl in [17], and more recently by Bradshaw et al. in [14], where they prove stronger lower bounds on the cop number of graphs with high girth. One deduces that the cop number of the Robertson graph is therefore at least 4. It is easily seen in Figure 1 that placing a cop on each of the three exterior vertices (a, b, c) only leaves 4 unprotected vertices, which form independent edges. ...
We show that the cop number of any graph on 18 or fewer vertices is at most 3. This answers a specific case of a question posed by Baird et al. on the minimum order of 4-cop-win graphs, first appearing in 2011. We also find all 3-cop-win graphs on 11 vertices, narrow down the possible 4-cop-win graphs on 19 vertices and get some progress on finding the minimum order of 3-cop-win planar graphs.
... Recently, the Meyniel's Conjecture on digraphs has received a lot of attention [3,4,6,7]. In this section we will use the same technique as above to find Eulerian digraphs of bounded degree and with arbitrarily large cop number. ...
The game of Cops and Robbers is a well known pursuit-evasion game played on graphs. It has been proved \cite{bounded_degree} that cubic graphs can have arbitrarily large cop number c(G), but the known constructions show only that the set is unbounded. In this paper we prove that there are arbitrarily large subcubic graphs G whose cop number is at least where . We also show that proving Meyniel's conjecture for graphs of bounded degree implies a weak Meyniel's conjecture for all graphs.
The game of Cops and Robbers is a well‐known pursuit‐evasion game played on graphs. It has been proved that cubic graphs can have arbitrarily large cop number c ( G ), but the known constructions show only that the set { c ( G ) ∣ G cubic } is unbounded. In this paper, we prove that there are arbitrarily large subcubic graphs G whose cop number is at least n 1 ∕ 2 − o ( 1 ) where n = ∣ V ( G ) ∣. We also show that proving Meyniel's conjecture for graphs of bounded degree implies a weaker version of Meyniel's conjecture for all graphs.
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 . That is, for every graph G on n vertices 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 . This proves that the 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 generators that are -free. This shows that the Kövári, Sós, and Turán theorem, stating that any -free graph of n vertices has at most edges, is tight up to a multiplicative constant even for abelian Cayley graphs.
In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant C, the cop number of every connected graph G is at most . In a separate paper, we showed that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph. The result was obtained by showing that the conjecture holds for a general class of graphs with some specific expansion‐type properties. In this paper, this deterministic result is used to show that the conjecture holds asymptotically almost surely for random d‐regular graphs when d = d(n) ≥ 3.
A large family of explicitk-regular Cayley graphsX is presented. These graphs satisfy a number of extremal combinatorial properties.
For eigenvaluesλ ofX eitherλ=±k or ¦λ¦≦2 √k−1. This property is optimal and leads to the best known explicit expander graphs.
The girth ofX is asymptotically ≧4/3 logk−1 ¦X¦ which gives larger girth than was previously known by explicit or non-explicit constructions.
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 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 . Comment: 4 pages
We show that the cop number of toroidal graphs is at most 3. This resolves a conjecture by Schroeder from 2001 which is implicit in a question by Andreae from 1986.
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).
"Categorical Perspectives" consists of introductory surveys as well as articles containing original research and complete proofs devoted mainly to the theoretical and foundational developments of category theory and its applications to other fields. A number of articles in the areas of topology, algebra and computer science reflect the varied interests of George Strecker to whom this work is dedicated. Notable also are an exposition of the contributions and importance of George Strecker's research and a survey chapter on general category theory. This work is an excellent reference text for researchers and graduate students in category theory and related areas.
Contributors: H.L. Bentley * G. Castellini * R. El Bashir * H. Herrlich * M. Husek * L. Janos * J. Koslowski * V.A. Lemin * A. Melton * G. Preuá * Y.T. Rhineghost * B.S.W. Schroeder * L. Schr"der * G.E. Strecker * A. Zmrzlina
We prove that the copnumber of a finite connected graph of genus g is bounded by [3/2g]+3. In particular this means that the copnumber of a toroidal graph is bounded by 4. We also sketch a proof that the copnumber of a graph of genus 2 is bounded by 5.
In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant C, the cop number of every connected graph G is at most . In this paper, we show that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph , which improves upon existing results showing that asymptotically almost surely the cop number of is provided that for some . We do this by first showing that the conjecture holds for a general class of graphs with some specific expansion-type properties. This will also be used in a separate paper on random d-regular graphs, where we show that the conjecture holds asymptotically almost surely when . © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015
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.).)
Concentrators are graphs used in the construction of switching networks that exhibit high connectivity. A technique for establishing the concentration properties of a graph by analysis of its eigenvalues is given. If the ratio of the subdominant eigenvalue to the dominant eigenvalue is small, the graph is a good concentrator. Generalized N-gons are very sparse, locally tree-like graphs for which the eigenvalues can be calculated with relative ease. The eigenvalue ratios for the known N-gons are calculated and show that the N-gons are excellent concentrators.
We consider the pursuit and evasion game on finite, connected, undirected graphs known as cops and robbers. Meyniel conjectured that for every graph on n vertices O(n 1 2) cops can win the game. We prove that this holds up to a log(n) factor for random graphs G(n, p) if p is not very small, and this is close to be tight unless the graph is very dense. We analyze the area- defending strategy (used by Aigner in case of planar graphs) and show examples where it can not be too efficient.
Recently, Bollobás, Janson and Riordan introduced a very general family of random graph models, producing inhomogeneous random graphs with Θ(n) edges. Roughly speaking, there is one model for each kernel, i.e., each symmetric measurable function from [0,1] 2 to the non-negative reals, although the details are much more complicated, to ensure the exact inclusion of many of the recent models for large-scale real-world networks. A different connection between kernels and random graphs arises in the recent work of Borgs, Chayes, Lovász, Sós, Szegedy and Vesztergombi. They introduced several natural metrics on dense graphs (graphs with n vertices and Θ(n 2) edges), showed that these metrics are equivalent, and gave a description of the completion of the space of all graphs with respect to any of these metrics in terms of graphons, which are essentially bounded kernels. One of the most appealing aspects of this work is the message that sequences of inhomogeneous quasi-random graphs are in a
Meyniel's conjecture is one of the deepest open problems on the cop num-ber of a graph. It states that for a connected graph G of order n, c(G) = O(√ n). While largely ignored for over 20 years, the conjecture is receiving increasing attention. We survey the origins of and recent developments towards the solution of the conjecture. We present some new results on so-called Meyniel extremal families containing graphs of order n satisfying c(G) ≥ d √ n, where d is a constant, and on the connections with the conjecture and the minimum order of connected k-cop-win graphs.
In this paper, we study the vertex pursuit game of Cops and Robbers where cops try to capture a robber on the vertices of the graph. The minimum number of cops required to win on a given graph G is the cop number of G. We present asymptotic results for the game of Cops and Robber played on a random graph G(n,p) for a wide range of p = p(n). It has been shown that the cop number as a function of an average degree forms an intriguing zigzag shape. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010
In a previous paper, Aigner and Fromme (Discrete Appl. Math.8 (1984), 1–12) considered a game played on a finite connected graph G where s pursuers try to catch one evader. They introduced c(G) as the minimal number s of pursuers that are sufficient to catch the evader and showed that, in general, c(G) can be arbitrarily high. On the other hand, they proved that c(G) ≤ 3 if G is planar. The present paper relates c(G) to the “forbidden minor concept.” Suppose that the graph H is not a minor of G and that, for a vertex hϵV(H) H − h has no isolated vertices. It is shown that this implies c(G) ≤ |E(H − h)|. As a consequence, one finds that, for each graph H, there exists a minimal positive integer α(H) such that c(G) ≤ α(H) when H is not a minor of G and, in addition, α(H) < |E(H)| for each connected H with at least two edges. These results are refined by proving that α(K5) = α(K3,3) = 3 and α(K5−) = α(K3,3−) = 2, thereby also refining the above result on planar graphs. (K5−(K3,3−) denotes K5(K3,3) minus an edge.) Further, , where Wn is the wheel with n rim vertices (n ≥ 3). In addition, we also establish an upper bound for c(G) in terms of the cross-cap number of G, thus providing a (partial) analogue to a result of Quilliot on the genus. Finally, the relationship between c(G) and simplicial decompositions is studied and a list of open problems is presented.
For X⊆V(G), let ∂X denote the set of edges of the graph G having one end in X and the other end in V(G)βX. The quantity , where the minimum is taken over all non-empty subsets X of V(G) with , is called the isoperimetric number of G. The basic properties of i(G) are discussed. Some upper and lower bounds on i(G) are derived, one in terms of |V(G)| and |E(G)| and two depending on the second smallest eigenvalue of the difference Laplacian matrix of G. The upper bound is a strong discrete version of the wellknown Cheeger inequality bounding the first eigenvalue of a Riemannian manifold. The growth and the diameter of a graph G are related to i(G). The isoperimetric number of Cartesian products of graphs is studied. Finally, regular graphs of fixed degree with large isoperimetric number are considered.
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.
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.
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.
The cop-number of a graph is the minimum number of cops needed to catch a robber on the graph, where the cops and the robber alternate moving from a vertex to a neighbouring vertex. It is conjectured by Meyniel that for a graph on n vertices O(root n) cops suffice. The aim of this paper is to investigate the cop-number of a random graph. We prove that for sparse random graphs the cop-number has order of magnitude n(1/2+o(1)). The best known strategy for general graphs is the area-defending strategy, where each cop 'controls' one region by himself. We show that, for general graphs, this strategy cannot be too effective: there are graphs that need at least n(1-o(1)) cops for this strategy.
F. Hasiri and I. Shinkar, Meyniel extremal families of Abelian Cayley graphs, Graphs Comb. 38 (2022). no. 3. <https://doi.org/10.1007/s00373-022-02460-8>
Algorithms and Combinatorics
- M. Molloy
- B. Reed
Problèmes de jeux de point fixe de connectivité et de représentation sur des graphes des ensembles ordonnés et des hypergraphes Ph
- A Quilliot