On Several Extremal Problems in Graph Theory Involving Gromov Hyperbolicity Constant
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Universidad Carlos III de Madrid On several extremal problems in graph theory involving Gromov Hyperbolicity Constant Author: Ver´onica Hern´andez Mart´ınez Departamento de Matem´aticas Advisor: Jos´eManuel Rodr´ıguez Garc´ıa Departamento de Matem´aticas Co-Advisor: Domingo de Guzm´an Pestana Galv´an Departamento de Matem´aticas October 25, 2017 Universidad Carlos III de Madrid On several extremal problems in graph theory involving Gromov Hyperbolicity Constant Author: Ver´onica Hern´andez Mart´ınez Department of Mathematics Advisor: Jos´eManuel Rodr´ıguez Garc´ıa Department of Mathematics Co-Advisor: Domingo de Guzm´an Pestana Galv´an Departamento de Matem´aticas Firma del Tribunal Calificador: Firma Presidente: Vocal: Secretario: Calificaci´on: Legan´es, de de 2017 to my family Agradecimientos / Acknowledgements En primer lugar quiero agradecer a mi director de tesis Jos´eManuel Rodr´ıguez Garc´ıa, cuyo gran apoyo y dedicaci´on han sido fundamentales, no solo para la realizaci´on de esta tesis, sino tambi´en para mi formaci´on como investigadora. Quiero agradecer a mi codirector de tesis Domingo de Guzm´an Pestana por sus consejos, paciencia e inter´es. Quiero agradecer al Departamento de Matem´aticas de la Universidad Carlos III de Madrid por la gran oportunidad que me brind´ode completar mi formaci´on como matem´atica. A los miembros del tribunal de esta disertaci´on doctoral, gracias por aceptar ser part´ıcipes en la defensa de esta tesis y por dedicarle su valioso tiempo a la lectura de este trabajo. Agradezco a mis padres, Julio y Magaly, por haber hecho posible que llegara hasta aqu´ı, por su amor y apoyo desde la distancia. Quiero expresar mi gratitud tambi´en a mis abuelos, mis t´ıas Susana y Bertica y a toda mi familia por su confianza en m´ı. Finalmente, a todos mis amigos que han estado siempre presentes, gracias por su gran apoyo moral y sus palabras de ´animo. ii Contents Introduction 6 1 A brief introduction to graph theory 12 1.1 Basicnotions ................................... 14 1.2 Subgraphs..................................... 16 1.3 Connectivityofgraphs .............................. 17 1.4 Somespecialgraphs ............................... 19 2 An introduction to Gromov hyperbolicity 23 2.1 Definition of Gromov Hyperbolicy and examples . 25 2.2 Gromov hyperbolicity, Mathematical Analysis and Geometry . 28 2.3 Main results on hyperbolic spaces . 30 2.4 Previous results on hyperbolic graphs . 32 3 On the order, size, and hyperbolicity constant 37 3.1 Bounds for B(n, m)................................ 38 3.2 Difference of the bounds of B(n, m)....................... 46 3.3 An inequality involving the diameter of a graph . 51 3.4 Computation of A(n, m)............................. 52 3.5 Non-simpleandweightedgraphs. 56 3.6 Randomgraphs.................................. 59 4 Diameter, minimum and maximum degree 61 4.1 On a classical Theorem on the diameter and minimum degree . 62 4.2 Upper bounds for b(n, δ0)............................. 71 4.3 Lower bounds and some precise values for b(n, δ0)............... 77 4.4 Computation of a(n, δ0) ............................. 83 4.5 Computation of β(n, ∆) ............................. 85 4.6 Computation of α(n, ∆) ............................. 88 2 5 Circumference, girth, and hyperbolicity constant 90 5.1 Technicalresults ................................. 91 5.2 Bounds for (g, c, n)............................... 98 A 5.3 Computation of (g, c, n)............................. 106 5.4 Bounds for A(g, c,B m)............................... 108 5.5 Computation of B(g, c, m)............................ 111 Conclusions 113 Open problems 116 Bibliography 118 List of Figures 1.1 ThegraphofthebridgesofK¨onigsberg. 12 1.2 Simple graph G1 and non-simple graph G2.................... 14 1.3 A subgraph G1 and an induced subgraph G2 of the graph G.......... 16 1.4 Representation of a connected graph G1 and a disconnected graph G2. 17 1.5 Pathgraphs..................................... 19 1.6 Cyclegraphs. ................................... 19 1.7 Completegraphs. ................................. 19 1.8 Completebipartitegraphs............................. 20 1.9 Stargraphs..................................... 20 1.10Wheelgraphs.................................... 21 2.1 R and R2 as examples of hyperbolic and non-hyperbolic spaces. 26 2.2 Any metric tree T verifies δ(T )=0........................ 26 2.3 First steps in order to compute the hyperbolicity constant of X. 27 2.4 Calculating the supremum over all geodesic triangles. 27 4 Introduction Gromov hyperbolicity was introduced by the Russian mathematician Mikhail Leonidovich Gromov in the setting of geometric group theory [59], [57], [41], but has played an increasing role in analysis on general metric spaces [19], [20], [8], with applications to the Martin boundary, invariant metrics in several complex variables [7] and extendability of Lipschitz mappings [79]. The concept of hyperbolicity appears also in discrete mathematics, algorithms and networking. For example, it has been shown empirically in [105] that the internet topology embeds with better accuracy into a hyperbolic space than into a Euclidean space of comparable dimension (formal proofs that the distortion is related to the hyperbolicity can be found in [109]); furthermore, it is evidenced that many real networks are hyperbolic (see, e.g., [1, 2, 40, 85]). Another important application of these spaces is the study of the spread of viruses through the internet (see [67, 68]). Furthermore, hyperbolic spaces are useful in secure transmission of information on the network (see [67, 68]). In [72] the authors study hyperbolicity in large scale networks (such as communication, citation, collaboration, peer- to-peer, friendship and other social networks) and propose that hyperbolicity, in conjunction with other local characteristics of networks, such as the degree distribution and clustering coefficients, provide a more complete unifying picture of networks, and helps classify in a parsimonious way what is otherwise a bewildering and complex array of features and characteristics specific to each natural and man-made network. The hyperbolicity has also been used extensively in the context of random graphs (see, e.g., [102, 103, 104]). In a geodesic metric space X, a geodesic triangle is the union of three geodesic in X. A geodesic metric space X is called hyperbolic (in the Gromov sense) if it satisfies the Rips condition, i.e. there is an upper bound of the distance of every point in a side of any geodesic triangle in X to the union of the two other sides (see Definition 2.1.1). The sharp upper bound in the previous definition is called the hyperbolicity constant of X and is denoted by δ(X). It is important to point out that there are several definitions of Gromov Hyperbolicity (see Chapter 2), although in this Thesis we work with Definition 2.1.1. For detailed expositions about Gromov hyperbolicity, see e.g. [3], [41], [57] or [108] (in this work we use the notations of [57]). In particular, graphs can be seen as geodesic metric spaces, considering that the points in a graph G are the vertices and, also, the points in the interior of any edge of G. The study of Gromov hyperbolic graphs is a subject of increasing interest in graph theory; see, e.g., [6, 16, 23, 28, 35, 37, 40, 42, 43, 54, 55, 67, 68, 75, 77, 81, 82, 85, 94, 107, 111, 113] and 6 7 the references therein. Last years several researchers have been interested in showing that metrics used in geo- metric function theory are Gromov hyperbolic. In particular, the equivalence of the hyper- bolicity of Riemannian manifolds and the hyperbolicity of a very simple graph was proved in [92, 94, 107], hence, it is useful to know hyperbolicity criteria for graphs. The main examples of hyperbolic graphs are trees. In fact, the hyperbolicity constant of a geodesic metric space can be viewed as a measure of how “tree-like” the space is, since those spaces X with δ(X) = 0 are precisely the metric trees. This is an interesting subject since, in many applications, one finds that the borderline between tractable and intractable cases may be the tree-like degree of the structure to be dealt with (see, e.g., [36]). For a finite graph with n vertices it is possible to compute δ(G) in time O(n3.69) [54] (this is improved in [40, 42]). Given a Cayley graph (of a presentation with solvable word problem) there is an algorithm which allows to decide if it is hyperbolic [89]. However, deciding whether or not a general infinite graph is hyperbolic is usually very difficult. Therefore, it is interesting to relate hyperbolicity with other properties of graphs. The papers [23, 113, 11, 30] prove, respectively, that chordal, k-chordal, edge-chordal and join graphs are hyperbolic. Moreover, in [11] it is shown that hyperbolic graphs are path-chordal graphs. These results relating chordality and hyperbolicity are improved in [81]. Some other authors have obtained results on hyperbolicity for particular classes of graphs: vertex-symmetric graphs, bipartite and intersection graphs, bridged graphs, expanders and median graphs [25, 43, 75, 80, 106]. The three main problems on Gromov hyperbolic graphs are the following: I. To obtain inequalities relating the hyperbolicity constant and other parameters of graphs. II. To study the hyperbolicity for important classes of graphs. III. To study the invariance of the hyperbolicity of graphs under appropriate transforma- tions. In this Thesis we study the extremal problems of maximazing and minimazing the hy- perbolicity constant on several families of graphs (problem II in the above list). In order to properly raise our research problem, we need to introduce some important definitions and make some remarks on the graphs we study. We consider simple (without loops or multiple edges) and connected graphs such that every edge has length 1. Note that to exclude multiple edges and loops is not an impor- tant loss of generality, since [16, Theorems 8 and 10] reduce the problem of computing the hyperbolicity constant of graphs with multiple edges and/or loops to the study of simple graphs.