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Graphs with small hyperbolicity constant
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Electronic Notes in Discrete Mathematics 46 (2014) 265–272 www.elsevier.com/locate/endm
Graphs with small hyperbolicity constant Sergio Bermudo 1 Pablo de Olavide University Carretera de Utrera Km. 1, 41013-Sevilla, Spain.
Jos´e M. Rodr´ıguez 2, Omar Rosario 3 Departamento de Matem´ aticas, Universidad Carlos III de Madrid Av. de la Universidad 30, 28911 Legan´es, Madrid, Spain.
Jos´e M. Sigarreta 4 Universidad Aut´ onoma de Guerrero Carlos E. Adame 5, Col. La Garita, Acapulco, Guerrero, M´exico.
Abstract If X is a geodesic metric space and x1 , x2 , x3 ∈ X, a geodesic triangle T = {x1 , x2 , x3 } is the union of the three geodesics [x1 x2 ], [x2 x3 ] and [x3 x1 ] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ(X) the sharpest hyperbolicity constant of X, i.e. δ(X) := inf{δ ≥ 0 : X is δ-hyperbolic }. In this paper we study the graphs with small hyperbolicity constant. Keywords: Graphs; infinite graphs; geodesics; Gromov hyperbolicity; hyperbolicity constant.
http://dx.doi.org/10.1016/j.endm.2014.08.035 1571-0653/© 2014 Published by Elsevier B.V.
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Introduction
Hyperbolic spaces play an important role in geometric group theory and in geometry of negatively curved spaces(see, e.g., [9]). The concept of Gromov hyperbolicity grasps the essence of negatively curved spaces like the classical hyperbolic space, Riemannian manifolds of negative sectional curvature, and of discrete spaces like trees and the Cayley graphs of many finitely generated groups. It is remarkable that a simple concept leads to such a rich general theory (see, e.g., [9]). The study of mathematical properties of Gromov hyperbolic spaces and its applications is a topic of recent and increasing interest in graph theory; see, for instance [1,2,4,6,7,8,10,11,12,13,15,16,17,18,19,20,22,23] and the references therein. In particular, in [16,21,22] it is proved the equivalence of the hyperbolicity of many negatively curved surfaces and the hyperbolicity of a very simple graph; hence, it is useful to know hyperbolicity criteria for graphs. The theory of Gromov’s spaces was used initially for the study of finitely generated groups, where it was demonstrated to have practical importance. This theory was applied principally to the study of automatic groups (see [14]), that play an important role in computer science. Another important application of these spaces is secure transmission of information by internet (see [10]), the spread of viruses through the network (see [10]), or the study of DNA data (see [6]). Now we give the basic facts about Gromov’s spaces. We say that a curve in a metric space (X, d) γ : [a, b] −→ X is a geodesic if it is an isometry, i.e., L(γ|[t,s] ) = d(γ(t), γ(s)) = |t − s| for every s, t ∈ [a, b]. We say that X is a geodesic metric space if for every x, y ∈ X there exists a geodesic joining x and y; we denote by [xy] any of such geodesics (since we do not require uniqueness of geodesics, this notation is ambiguous, but it is convenient). It is clear that every geodesic metric space is path-connected. If X is a graph, we use the notation [u, v] for the edge of a graph joining the vertices u and v. We consider graphs such that the length of every edge is k ∈ (0, ∞). In order to consider a graph G as a geodesic metric space, we must identify any edge [u, v] ∈ E(G) with the real interval [0, k]; therefore, any point in the interior of any edge is a point of G. Hence, if we consider the edge [u, v] as a graph with just one edge, then it is isometric to [0, k]. A connected graph G 1 2 3 4
Email: Email: Email: Email:
[email protected] [email protected] [email protected] [email protected]
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is naturally equipped with a distance defined on its points, induced by taking shortest paths in G. Then, we see G as a metric graph. Along the paper we just consider simple (without loops and multiple edges) connected graphs whose edges have length equal to k; these properties guarantee that the graphs are geodesic metric spaces. Note that to exclude multiple edges and loops is not an important loss of generality, since [1, Theorems 8 and 10] reduce the problem of compute the hyperbolicity constant of graphs with multiple edges and/or loops to the study of simple graphs. If X is a geodesic metric space and J = {J1 , J2 , . . . , Jn }, with Jj ⊆ X, we say that J is δ-thin if for every x ∈ Ji we have that d(x, ∪j6=i Jj ) ≤ δ. We denote by δ(J) the sharpest thin constant of J, i.e., δ(J) := inf{δ ≥ 0 : J is δ-thin }. If x1 , x2 , x3 ∈ X, a geodesic triangle T = {x1 , x2 , x3 } is the union of the three geodesics [x1 x2 ], [x2 x3 ] and [x3 x1 ]; it is usual to write also T = {[x1 x2 ], [x2 x3 ], [x3 x1 ]} and we will say that x1 , x2 , x3 are the vertices of the triangle. The space X is δ-hyperbolic (or satisfies the Rips condition with constant δ) if every geodesic triangle in X is δ-thin. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) := sup{δ(T ) : T is a geodesic triangle in X }. We say that X is hyperbolic if X is δ-hyperbolic for some δ ≥ 0. If X is hyperbolic, then δ(X) = inf{δ ≥ 0 : X is δ-hyperbolic }. Note that every geodesic polygon with n sides in a δ-hyperbolic space is (n − 2)δ-thin. Since the hyperbolicity of many geodesic metric spaces is equivalent to the hyperbolicity of some graphs related to them (see, e.g., [5]), the study of hyperbolic graphs becomes an interesting topic. The main aim of this paper is to study the graphs with small hyperbolicity constant, i.e., the graphs which are like trees (in the Gromov sense). We state now the main results in this work and the lemmas used in their proofs, see [3] for the detailed proofs.
2
Previous results
As usual, by cycle we mean a simple closed curve, i.e., a path with different vertices, unless the last one, which is equal to the first vertex. It is known (see [21, Lemma 2.1]) that, for every graph G, it is satisfied δ(G) = sup{δ(T ) : T is a geodesic triangle in G that is a cycle}. Let G be a graph with edges of the same length k. We denote by J(G) the union of the set V (G) and the midpoints of the edges of G. Consider the
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set T1 of geodesic triangles T in G that are cycles and such that the three vertices of the triangle T belong to J(G), and denote by δ1 (G) the infimum of the constants λ such that every triangle in T1 is λ-thin. The following results, which appear in [4, Theorems 2.5, 2.6 and 2.7], will be used throughout the paper. Theorem 2.1 For every graph G with edges of lengths k we have δ1 (G) = δ(G). Theorem 2.2 For every hyperbolic graph G with edges of lengths k, δ(G) is a multiple of k4 . The following result is a direct consequence of Theorems 2.1 and 2.2; it states that in the hyperbolic graphs with edges of length k there always exists a geodesic triangle for which the hyperbolicity constant is attained. Theorem 2.3 For any hyperbolic graph G with edges of lengths k, there exists a geodesic triangle T ∈ T1 such that δ(T ) = δ(G).
3
Characterizations of graphs with small hyperbolicity constant
The results in this section show some characterizations for hyperbolic graphs with small hyperbolicity constant. In this sense, in [12, Theorem 11] appears the following result. Theorem 3.1 Let G be any graph with edges of length k. (a) δ(G) = 0 if and only if G is a tree. (b) δ(G) = k4 , k2 is not satisfied for any graph G. (c) δ(G) =
3k 4
if and only if G is not a tree and every cycle in G has length 3k.
In order to characterize the graphs with edges of length k and hyperbolicity constant greater than 3k it is necessary to obtain some previous results. If 4 H is a subgraph of G and w ∈ V (H), we denote by degH (w) the degree of the vertex w in the subgraph induced by V (H). A subgraph H of G is said isometric if dH (x, y) = dG (x, y) for every x, y ∈ H. Note that this condition is equivalent to dH (u, v) = dG (u, v) for every vertices u, v ∈ V (H). Theorem 3.2 Let G be any graph with edges of length k. Then δ(G) ≥ 5k 4 if and only if there exist a cycle g in G with length L(g) ≥ 5k and a vertex
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w ∈ g such that degg (w) = 2. For every m ≥ 4, we say that a graph G with edges of length 1 is m-chordal (see [23]) if for any cycle C in G with length L(C) ≥ m, there exists an edge joining two non-consecutive vertices x, y of C. Given a cycle C in G, we say that a geodesic g = [uv] is a shortcut if u, v ∈ V (C), L(g) = d(u, v) < dC (u, v) and g ∩ C = {u, v}. Corollary 3.3 Let G be any graph with edges of length 1. If δ(G) ≤ 1, then G is 5-chordal. Proposition 3.4 Let G be any graph with edges of length k. If δ(G) = k, then G has a cycle isomorphic to C4 . Lemma 3.5 Let G be a graph with edges of length 1 such that δ(G) = 45 . If G contains a cycle C of length 6 and there exist x, y ∈ C such that d(x, y) = 3, then G has a cycle isomorphic to C5 . We will need the following results (see [1, Lemma 9] and [20, Theorem 11]) in order to prove Proposition 3.9 below. Lemma 3.6 If H is an isometric subgraph of G, then δ(H) ≤ δ(G). Lemma 3.7 If Cn is the cycle graph with n vertices and edges of length k, . then δ(Cn ) = nk 4 Corollary 3.8 Let G be any graph with edges of length k. If G contains an . isometric subgraph which is isomorphic to Cn , then δ(G) ≥ nk 4 Proposition 3.9 Let G be any graph with edges of length k. If δ(G) = then G has a cycle isomorphic to C5 .
5k , 4
Looking at Propositions 3.4 and 3.9 it seems logical to think that, if δ(G) = then G has a cycle isomorphic to C6 or, more generally, if δ(G) = nk , then 4 G has a cycle isomorphic to Cn for each n > 5. However, for each n > 5 we have found a graph Gn such that δ(Gn ) = nk and Gn does not have any cycle 4 isomorphic to Cn . 6k , 4
Theorem 3.10 Let G be any graph with edges of length k. Then δ(G) = k if and only if the following conditions hold: (1) There exists a cycle isomorphic to C4 . (2) For every cycle γ such that L(γ) ≥ 5k and for every vertex w ∈ γ, it is satisfied degγ (w) ≥ 3.
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Proposition 3.11 Let G be any graph with edges of length k. Assume that the following conditions hold: (1) There exist a cycle g in G such that L(g) ≥ 5k and a vertex w ∈ g satisfying degg (w) = 2. (2) For every cycle γ such that L(γ) ≥ 6k, we have diam(γ) ≤ Then we have δ(G) =
5k . 2
5k . 4
Proposition 3.12 Let G be any graph with edges of length k. If δ(G) ≥ then there exists a cycle g in G such that L(g) ≥ 6k and diam(g) ≥ 3k.
4
3k , 2
Hyperbolicity constant and effective diameter
A graph with small hyperbolicity constant can have arbitrarily large diameter: the path graph with n vertices Pn verifies δ(Pn ) = 0 and diam(Pn ) = diam V (Pn ) = n − 1 for every n. However, there is a concept related with the diameter, the effective diameter, which is small when the hyperbolicity constant is small. We say that a vertex v of a graph G is a cut-vertex if G \ {v} is not connected. A graph is two-connected if it is connected and it does not contain cut-vertices. Given any edge in G, let us consider the maximal two-connected subgraph containing it. We call to the set of these maximal two-connected subgraphs {Gn }n the canonical T-decomposition of G. Given a graph G and its canonical T-decomposition {Gn }, we define the effective diameter as effdiam V (G) := sup diam V (Gn ), n
effdiam (G) := sup diam (Gn ). n
We have the following result. Proposition 4.1 Let G be any graph with edges of length k. If δ(G) < k, . then effdiamV (G) = k and effdiam(G) ≤ 3k 2 We need the following result in order to prove now a bound for effdiamV (G) when δ(G) = k. Proposition 4.2 Let G be any graph with edges of length k. If G does not have cut-vertices and δ(G) ≤ k, then diamV (G) ≤ 2k. Finally, we obtain an upper bound of effdiamV (G) for every graph G with δ(G) = k.
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Theorem 4.3 Let G be any graph with edges of length k. If δ(G) = k, then effdiamV (G) ≤ 2k. : Let G be Remark 4.4 It is not possible to bound effdiamV (G) if δ(G) ≥ 3k 2 the Cayley graph of the group Z × Z2 (G has the shape of an infinite railway). We have δ(G) = 3k and the canonical T-decomposition of G has just a graph 2 G1 = G; hence, effdiamV (G) =diamV (G1 ) = ∞.
References [1] Bermudo, S., J. M. Rodr´ıguez, J. M. Sigarreta, and J.-M. Vilaire, Gromov hyperbolic graphs, Discrete Math. 313 (15) (2013), 1575–1585. [2] Bermudo, S., J. M. Rodr´ıguez, J. M. Sigarreta, and E. Touris, Hyperbolicity and complement of graphs, Appl. Math. Letters 24 (2011), 1882-1887. [3] Bermudo, S., J. M. Rodr´ıguez, O. Rosario, and J. M. Sigarreta, Small values of the hyperbolicity constant in graphs. Submited. [4] Bermudo, S., J. M. Rodr´ıguez, and J. M. Sigarreta, Computing the hyperbolicity constant, Comput. Math. Appl. 62 (2011), 4592-4595. [5] Bowditch, B. H., “Notes on Gromov’s hyperbolicity criterion for path-metric spaces”, Group theory from a geometrical viewpoint, Trieste, 1990 (ed. E. Ghys, A. Haefliger and A. Verjovsky; World Scientific, River Edge, NJ, 1991) 64-167. [6] Brinkmann, G., J. Koolen, and V. Moulton, On the hyperbolicity of chordal graphs, Ann. Comb. 5 (2001), 61-69. [7] Carballosa, W., D. Pestana, J. M. Rodr´ıguez, and J. M. Sigarreta, Distortion of the hyperbolicity constant of a graph, Electr. J. Combin. 19 (2012), P67. [8] Carballosa, W., J. M. Rodr´ıguez, J. M. Sigarreta, and M. Villeta, On the hyperbolicity constant of line graphs, Electr. J. Combin. 18 (2011), P210. [9] Ghys, E., and P. de la Harpe, “Sur les Groupes Hyperboliques d’apr`es Mikhael Gromov”, Progress in Mathematics 83, Birkhauser Boston Inc., Boston, MA., (1990). [10] Jonckheere, E. A., and P. Lohsoonthorn, Geometry of network security, American Control Conference, ACC (2004), 111-151. [11] Koolen, J. H., and V. Moulton, Hyperbolic Bridged Graphs, Europ. J. Combin. 23 (2002), 683-699.
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[12] Michel, J., J. M. Rodr´ıguez, J. M. Sigarreta, and M. Villeta, Hyperbolicity and parameters of graphs, Ars Combin. 100 (2011), 43-63. [13] Michel, J., J. M. Rodr´ıguez, J. M. Sigarreta, and M. Villeta, Gromov hyperbolicity in cartesian product graphs, Proc. Indian Acad. Sci. (Math. Sci.) 120 (2010), 1-17. [14] Oshika, K., “Discrete groups”, AMS Bookstore, 2002. [15] Pestana, D., J. M. Rodr´ıguez, J. M. Sigarreta, and M. Villeta, em Gromov hyperbolic cubic graphs, Central Europ. J. Math. 10 (2012), 1141-1151. [16] Portilla, A., J. M. Rodr´ıguez, and E. Tour´ıs, Gromov hyperbolicity through decomposition of metric spaces II, J. Geom. Anal. 14 (2004), 123-149. [17] Portilla, A., J. M. Rodr´ıguez, and E. Tour´ıs, Stability of Gromov hyperbolicity, J. Adv. Math. Studies 2 (2009), No. 2, 77-96. [18] Portilla, A., and E. Tour´ıs, A characterization of Gromov hyperbolicity of surfaces with variable negative curvature, Publ. Mat. 53 (2009), 83-110. [19] Rodr´ıguez, J. M., and J. M. Sigarreta, Bounds on Gromov hyperbolicity constant in graphs, Proc. Indian Acad. Sci. (Math. Sci.) 122 (2012), 53-65. [20] Rodr´ıguez, J. M., J. M. Sigarreta, J.-M. Vilaire, and M. Villeta, On the hyperbolicity constant in graphs, Discrete Math. 311 (2011), 211-219. [21] Rodr´ıguez, J. M., and E. Tour´ıs, Gromov hyperbolicity through decomposition of metric spaces, Acta Math. Hung. 103 (2004), 53-84. [22] Tour´ıs, E., Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces, J. Math. Anal. Appl. 380 (2011), 865-881. [23] Wu, Y., and C. Zhang, Chordality and hyperbolicity of a graph, Electr. J. Combin. 18 (2011), P43.
Electronic Notes in Discrete Mathematics 46 (2014) 265–272 www.elsevier.com/locate/endm
Graphs with small hyperbolicity constant Sergio Bermudo 1 Pablo de Olavide University Carretera de Utrera Km. 1, 41013-Sevilla, Spain.
Jos´e M. Rodr´ıguez 2, Omar Rosario 3 Departamento de Matem´ aticas, Universidad Carlos III de Madrid Av. de la Universidad 30, 28911 Legan´es, Madrid, Spain.
Jos´e M. Sigarreta 4 Universidad Aut´ onoma de Guerrero Carlos E. Adame 5, Col. La Garita, Acapulco, Guerrero, M´exico.
Abstract If X is a geodesic metric space and x1 , x2 , x3 ∈ X, a geodesic triangle T = {x1 , x2 , x3 } is the union of the three geodesics [x1 x2 ], [x2 x3 ] and [x3 x1 ] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. We denote by δ(X) the sharpest hyperbolicity constant of X, i.e. δ(X) := inf{δ ≥ 0 : X is δ-hyperbolic }. In this paper we study the graphs with small hyperbolicity constant. Keywords: Graphs; infinite graphs; geodesics; Gromov hyperbolicity; hyperbolicity constant.
http://dx.doi.org/10.1016/j.endm.2014.08.035 1571-0653/© 2014 Published by Elsevier B.V.
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Introduction
Hyperbolic spaces play an important role in geometric group theory and in geometry of negatively curved spaces(see, e.g., [9]). The concept of Gromov hyperbolicity grasps the essence of negatively curved spaces like the classical hyperbolic space, Riemannian manifolds of negative sectional curvature, and of discrete spaces like trees and the Cayley graphs of many finitely generated groups. It is remarkable that a simple concept leads to such a rich general theory (see, e.g., [9]). The study of mathematical properties of Gromov hyperbolic spaces and its applications is a topic of recent and increasing interest in graph theory; see, for instance [1,2,4,6,7,8,10,11,12,13,15,16,17,18,19,20,22,23] and the references therein. In particular, in [16,21,22] it is proved the equivalence of the hyperbolicity of many negatively curved surfaces and the hyperbolicity of a very simple graph; hence, it is useful to know hyperbolicity criteria for graphs. The theory of Gromov’s spaces was used initially for the study of finitely generated groups, where it was demonstrated to have practical importance. This theory was applied principally to the study of automatic groups (see [14]), that play an important role in computer science. Another important application of these spaces is secure transmission of information by internet (see [10]), the spread of viruses through the network (see [10]), or the study of DNA data (see [6]). Now we give the basic facts about Gromov’s spaces. We say that a curve in a metric space (X, d) γ : [a, b] −→ X is a geodesic if it is an isometry, i.e., L(γ|[t,s] ) = d(γ(t), γ(s)) = |t − s| for every s, t ∈ [a, b]. We say that X is a geodesic metric space if for every x, y ∈ X there exists a geodesic joining x and y; we denote by [xy] any of such geodesics (since we do not require uniqueness of geodesics, this notation is ambiguous, but it is convenient). It is clear that every geodesic metric space is path-connected. If X is a graph, we use the notation [u, v] for the edge of a graph joining the vertices u and v. We consider graphs such that the length of every edge is k ∈ (0, ∞). In order to consider a graph G as a geodesic metric space, we must identify any edge [u, v] ∈ E(G) with the real interval [0, k]; therefore, any point in the interior of any edge is a point of G. Hence, if we consider the edge [u, v] as a graph with just one edge, then it is isometric to [0, k]. A connected graph G 1 2 3 4
Email: Email: Email: Email:
[email protected] [email protected] [email protected] [email protected]
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is naturally equipped with a distance defined on its points, induced by taking shortest paths in G. Then, we see G as a metric graph. Along the paper we just consider simple (without loops and multiple edges) connected graphs whose edges have length equal to k; these properties guarantee that the graphs are geodesic metric spaces. Note that to exclude multiple edges and loops is not an important loss of generality, since [1, Theorems 8 and 10] reduce the problem of compute the hyperbolicity constant of graphs with multiple edges and/or loops to the study of simple graphs. If X is a geodesic metric space and J = {J1 , J2 , . . . , Jn }, with Jj ⊆ X, we say that J is δ-thin if for every x ∈ Ji we have that d(x, ∪j6=i Jj ) ≤ δ. We denote by δ(J) the sharpest thin constant of J, i.e., δ(J) := inf{δ ≥ 0 : J is δ-thin }. If x1 , x2 , x3 ∈ X, a geodesic triangle T = {x1 , x2 , x3 } is the union of the three geodesics [x1 x2 ], [x2 x3 ] and [x3 x1 ]; it is usual to write also T = {[x1 x2 ], [x2 x3 ], [x3 x1 ]} and we will say that x1 , x2 , x3 are the vertices of the triangle. The space X is δ-hyperbolic (or satisfies the Rips condition with constant δ) if every geodesic triangle in X is δ-thin. We denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) := sup{δ(T ) : T is a geodesic triangle in X }. We say that X is hyperbolic if X is δ-hyperbolic for some δ ≥ 0. If X is hyperbolic, then δ(X) = inf{δ ≥ 0 : X is δ-hyperbolic }. Note that every geodesic polygon with n sides in a δ-hyperbolic space is (n − 2)δ-thin. Since the hyperbolicity of many geodesic metric spaces is equivalent to the hyperbolicity of some graphs related to them (see, e.g., [5]), the study of hyperbolic graphs becomes an interesting topic. The main aim of this paper is to study the graphs with small hyperbolicity constant, i.e., the graphs which are like trees (in the Gromov sense). We state now the main results in this work and the lemmas used in their proofs, see [3] for the detailed proofs.
2
Previous results
As usual, by cycle we mean a simple closed curve, i.e., a path with different vertices, unless the last one, which is equal to the first vertex. It is known (see [21, Lemma 2.1]) that, for every graph G, it is satisfied δ(G) = sup{δ(T ) : T is a geodesic triangle in G that is a cycle}. Let G be a graph with edges of the same length k. We denote by J(G) the union of the set V (G) and the midpoints of the edges of G. Consider the
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set T1 of geodesic triangles T in G that are cycles and such that the three vertices of the triangle T belong to J(G), and denote by δ1 (G) the infimum of the constants λ such that every triangle in T1 is λ-thin. The following results, which appear in [4, Theorems 2.5, 2.6 and 2.7], will be used throughout the paper. Theorem 2.1 For every graph G with edges of lengths k we have δ1 (G) = δ(G). Theorem 2.2 For every hyperbolic graph G with edges of lengths k, δ(G) is a multiple of k4 . The following result is a direct consequence of Theorems 2.1 and 2.2; it states that in the hyperbolic graphs with edges of length k there always exists a geodesic triangle for which the hyperbolicity constant is attained. Theorem 2.3 For any hyperbolic graph G with edges of lengths k, there exists a geodesic triangle T ∈ T1 such that δ(T ) = δ(G).
3
Characterizations of graphs with small hyperbolicity constant
The results in this section show some characterizations for hyperbolic graphs with small hyperbolicity constant. In this sense, in [12, Theorem 11] appears the following result. Theorem 3.1 Let G be any graph with edges of length k. (a) δ(G) = 0 if and only if G is a tree. (b) δ(G) = k4 , k2 is not satisfied for any graph G. (c) δ(G) =
3k 4
if and only if G is not a tree and every cycle in G has length 3k.
In order to characterize the graphs with edges of length k and hyperbolicity constant greater than 3k it is necessary to obtain some previous results. If 4 H is a subgraph of G and w ∈ V (H), we denote by degH (w) the degree of the vertex w in the subgraph induced by V (H). A subgraph H of G is said isometric if dH (x, y) = dG (x, y) for every x, y ∈ H. Note that this condition is equivalent to dH (u, v) = dG (u, v) for every vertices u, v ∈ V (H). Theorem 3.2 Let G be any graph with edges of length k. Then δ(G) ≥ 5k 4 if and only if there exist a cycle g in G with length L(g) ≥ 5k and a vertex
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w ∈ g such that degg (w) = 2. For every m ≥ 4, we say that a graph G with edges of length 1 is m-chordal (see [23]) if for any cycle C in G with length L(C) ≥ m, there exists an edge joining two non-consecutive vertices x, y of C. Given a cycle C in G, we say that a geodesic g = [uv] is a shortcut if u, v ∈ V (C), L(g) = d(u, v) < dC (u, v) and g ∩ C = {u, v}. Corollary 3.3 Let G be any graph with edges of length 1. If δ(G) ≤ 1, then G is 5-chordal. Proposition 3.4 Let G be any graph with edges of length k. If δ(G) = k, then G has a cycle isomorphic to C4 . Lemma 3.5 Let G be a graph with edges of length 1 such that δ(G) = 45 . If G contains a cycle C of length 6 and there exist x, y ∈ C such that d(x, y) = 3, then G has a cycle isomorphic to C5 . We will need the following results (see [1, Lemma 9] and [20, Theorem 11]) in order to prove Proposition 3.9 below. Lemma 3.6 If H is an isometric subgraph of G, then δ(H) ≤ δ(G). Lemma 3.7 If Cn is the cycle graph with n vertices and edges of length k, . then δ(Cn ) = nk 4 Corollary 3.8 Let G be any graph with edges of length k. If G contains an . isometric subgraph which is isomorphic to Cn , then δ(G) ≥ nk 4 Proposition 3.9 Let G be any graph with edges of length k. If δ(G) = then G has a cycle isomorphic to C5 .
5k , 4
Looking at Propositions 3.4 and 3.9 it seems logical to think that, if δ(G) = then G has a cycle isomorphic to C6 or, more generally, if δ(G) = nk , then 4 G has a cycle isomorphic to Cn for each n > 5. However, for each n > 5 we have found a graph Gn such that δ(Gn ) = nk and Gn does not have any cycle 4 isomorphic to Cn . 6k , 4
Theorem 3.10 Let G be any graph with edges of length k. Then δ(G) = k if and only if the following conditions hold: (1) There exists a cycle isomorphic to C4 . (2) For every cycle γ such that L(γ) ≥ 5k and for every vertex w ∈ γ, it is satisfied degγ (w) ≥ 3.
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Proposition 3.11 Let G be any graph with edges of length k. Assume that the following conditions hold: (1) There exist a cycle g in G such that L(g) ≥ 5k and a vertex w ∈ g satisfying degg (w) = 2. (2) For every cycle γ such that L(γ) ≥ 6k, we have diam(γ) ≤ Then we have δ(G) =
5k . 2
5k . 4
Proposition 3.12 Let G be any graph with edges of length k. If δ(G) ≥ then there exists a cycle g in G such that L(g) ≥ 6k and diam(g) ≥ 3k.
4
3k , 2
Hyperbolicity constant and effective diameter
A graph with small hyperbolicity constant can have arbitrarily large diameter: the path graph with n vertices Pn verifies δ(Pn ) = 0 and diam(Pn ) = diam V (Pn ) = n − 1 for every n. However, there is a concept related with the diameter, the effective diameter, which is small when the hyperbolicity constant is small. We say that a vertex v of a graph G is a cut-vertex if G \ {v} is not connected. A graph is two-connected if it is connected and it does not contain cut-vertices. Given any edge in G, let us consider the maximal two-connected subgraph containing it. We call to the set of these maximal two-connected subgraphs {Gn }n the canonical T-decomposition of G. Given a graph G and its canonical T-decomposition {Gn }, we define the effective diameter as effdiam V (G) := sup diam V (Gn ), n
effdiam (G) := sup diam (Gn ). n
We have the following result. Proposition 4.1 Let G be any graph with edges of length k. If δ(G) < k, . then effdiamV (G) = k and effdiam(G) ≤ 3k 2 We need the following result in order to prove now a bound for effdiamV (G) when δ(G) = k. Proposition 4.2 Let G be any graph with edges of length k. If G does not have cut-vertices and δ(G) ≤ k, then diamV (G) ≤ 2k. Finally, we obtain an upper bound of effdiamV (G) for every graph G with δ(G) = k.
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Theorem 4.3 Let G be any graph with edges of length k. If δ(G) = k, then effdiamV (G) ≤ 2k. : Let G be Remark 4.4 It is not possible to bound effdiamV (G) if δ(G) ≥ 3k 2 the Cayley graph of the group Z × Z2 (G has the shape of an infinite railway). We have δ(G) = 3k and the canonical T-decomposition of G has just a graph 2 G1 = G; hence, effdiamV (G) =diamV (G1 ) = ∞.
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