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Symmetry 2018, 10(7), 279; https://doi.org/10.3390/sym10070279

Article
Hyperbolicity of Direct Products of Graphs
1
Department of Mathematics and Statistics, Florida International University, 11200 SW 8th Street, Miami, FL 33199, USA
2
Department of Mathematics, Miami Dade College, 300 NE Second Ave. Miami, FL 33132, USA
3
Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, Leganés, 28911 Madrid, Spain
4
Facultad CC. Sociales de Talavera, Universidad de Castilla La Mancha, Avda. Real Fábrica de Seda, s.n. Talavera de la Reina, 45600 Toledo, Spain
*
Author to whom correspondence should be addressed.
Received: 11 June 2018 / Accepted: 9 July 2018 / Published: 12 July 2018

Abstract

:
It is well-known that the different products of graphs are some of the more symmetric classes of graphs. Since we are interested in hyperbolicity, it is interesting to study this property in products of graphs. Some previous works characterize the hyperbolicity of several types of product graphs (Cartesian, strong, join, corona and lexicographic products). However, the problem with the direct product is more complicated. The symmetry of this product allows us to prove that, if the direct product G 1 × G 2 is hyperbolic, then one factor is bounded and the other one is hyperbolic. Besides, we prove that this necessary condition is also sufficient in many cases. In other cases, we find (not so simple) characterizations of hyperbolic direct products. Furthermore, we obtain good bounds, and even formulas in many cases, for the hyperbolicity constant of the direct product of some important graphs (as products of path, cycle and even general bipartite graphs).
Keywords:
direct product of graphs; geodesics; Gromov hyperbolicity; bipartite graphs

1. Introduction

An interesting topic in graph theory is the study of the different types of products of graphs [1]. In particular, given two graphs G 1 , G 2 , the direct product G 1 × G 2 is defined as the graph with vertices the (Cartesian) product of V ( G 1 ) and V ( G 2 ) , and two vertices ( u 1 , v 1 ) , ( u 2 , v 2 ) V ( G 1 × G 2 ) are connected by an edge if and only if [ u 1 , u 2 ] E ( G 1 ) and [ v 1 , v 2 ] E ( G 2 ) . The direct product is associative and commutative. Direct product was introduced in Principia Mathematica by Russell and Whitehead.
Weichsel observed that G 1 × G 2 is connected if and only if the graphs G 1 and G 2 are connected and G 1 or G 2 is not a bipartite graph [2], i.e., there exists an odd cycle. The direct product is known with different names: tensor product, conjunction, categorical product, Kronecker product and cardinal product. There are many works studying several properties of direct products. These works include structural results [3,4,5,6,7,8], hamiltonian properties [9,10], and above all the well-known Hedetniemi’s conjecture (see [11,12]). Imrich has an algorithm in [13] which can recognize in polynomial time if a graph is a direct product; furthermore, the algorithm provides a factorization if the graph is a direct product. This fact facilitates the computational use of the direct product of graphs.
Hyperbolic spaces are an important tool in geometry and group theory [14,15,16]. Gromov hyperbolicity is a meeting point for different spaces: some of them continuous (hyperbolic plane and many Riemannian manifolds with negative curvature) and some of them discrete (trees and many graphs) [14,15,16].
Gromov hyperbolicity was introduced in the context of finitely generated groups [16], and it was applied, in the science of computation, to the study of automatic groups [17,18]. Gromov hyperbolicity is useful in networking, algorithms and discrete mathematics [19,20,21,22,23,24]; also, many real networks are hyperbolic [25,26,27,28,29]. Besides, there are several important applications of hyperbolic spaces to the Internet [30,31,32,33,34] and to random graphs [35,36,37]. It has recently been pointed out that also some aspects of biological systems require hyperbolicity for proper functioning [38]. In [39], it was proven that, for a large class of Riemannian surfaces endowed with a metric of negative curvature, there is a very simple graph related with the surface such that the surface is hyperbolic if and only if the graph is hyperbolic; therefore, it is interesting to study hyperbolic graphs to understand hyperbolic surfaces.
All these facts show the increasing interest of hyperbolic graphs (see, e.g., [19,24,25,26,27,32,33,35,36,37,39,40,41,42,43,44,45,46,47] and the references therein).
In this paper, let us denote by G = ( V , E ) = ( V ( G ) , E ( G ) ) a connected graph with V ( G ) . We consider that the length of each edge is 1. In addition, we assume that the graph does not have either multiple edges or loops.
Trees are the graphs with hyperbolicity constant zero. Thus, we can view the hyperbolicity constant as a measure of how “tree-like” the space is. This is an important subject (see, e.g., [48,49]).
From a computational viewpoint, we can obtain δ ( G ) in time O ( n 3.69 ) for graphs with n vertices [50]. In addition, there is an algorithm which decides if a Cayley graph is hyperbolic [51]. In [52], this algorithm is improved, allowing to obtain δ ( G ) in time O ( n 2 ) , but only if the graph is given in terms of its distance-matrix. However, it is usually very difficult to decide if an infinite graph is hyperbolic. Therefore, it is useful to study hyperbolicity for particular classes of graphs. There are many works dealing with the hyperbolicity of different types of graphs: median graphs [53], line graphs [54,55,56], cubic graphs [57], complement graphs [58], regular graphs [59], chordal graphs [25,42,45,60], planar graphs [61,62], bipartite and intersection graphs [63], vertex-symmetric graphs [64], periodic graphs [65,66], expanders [34], bridged graphs [67], short graphs [68], graph minors [69], graphs with small hyperbolicity constant [70], Mycielskian graphs [71], geometric graphs [56,72], and some types of products of graphs: Cartesian product and sum [46,73], strong product [74], lexicographic product [75], and corona and join product [76].
Some of these works give results about the hyperbolicity of some unary operations in graphs:
A line graph is hyperbolic if and only if the original graph does [54,55,56].
For a large class of minor graphs, the minor graph is hyperbolic if and only if the original graph does [69].
Mycielskian graphs are always hyperbolic [71].
Now, we summarize the known results about the hyperbolicity of the main class of binary operations in graphs: products of graphs.
The Cartesian product is hyperbolic if and only if one factor graph is bounded and the other one is hyperbolic [46].
The same holds for the strong product [74].
The corona product G 1 G 2 is hyperbolic if and only if the first factor G 1 is hyperbolic, and the join G 1 G 2 is always hyperbolic [76].
The Cartesian sum G 1 G 2 is always hyperbolic, if the factors have at least two vertices [73].
The lexicographic product graph G 1 G 2 is hyperbolic if and only if G 1 does, if the first factor has at least two vertices [75].
The goal of this paper is the characterization in many cases of the direct product of graphs which are hyperbolic. Here, the situation is more complicated than with other products of graphs. This is partly because the direct product of two bipartite graphs (i.e., graphs without odd cycles) is already disconnected and the formula for the distance in G 1 × G 2 is more complicated that in the case of other products of graphs. The symmetry of this product allows us to show that, if G 1 × G 2 is hyperbolic, then one factor is hyperbolic and the other one is bounded (see Theorem 10). Besides, we prove that this necessary condition is also sufficient in many cases. If G 1 is a hyperbolic graph and G 2 is a bounded graph, then we prove that G 1 × G 2 is hyperbolic when G 2 has some odd cycle (Theorem 3) or G 1 and G 2 do not have odd cycles (Theorem 4). One could think that otherwise (if G 1 has some odd cycle and G 2 does not have odd cycles) this necessary condition is also sufficient; however, Theorem 15 allows constructing in an easy way examples G 1 , G 2 (with G 1 hyperbolic and G 2 bounded) such that G 1 × G 2 is not hyperbolic. This shows that the characterization of hyperbolic direct products is a more difficult task when G 1 has some odd cycle and G 2 does not have odd cycles. Theorems 11 and 12 provide sufficient conditions for non-hyperbolicity and hyperbolicity, respectively. Besides, Theorems 15 and Corollary 5 characterize the hyperbolicity of G 1 × G 2 under some additional conditions. Furthermore, we obtain good bounds, and even formulas in many cases, for the hyperbolicity constant of the direct product of some important graphs; in particular, Theorem 18 provides the hyperbolicity constant of many direct products of bipartite graphs, and Theorems 17 and 19 give the hyperbolicity constant of many direct products of path and cycle graphs.
We want to remark that, in a general context, the hypothesis on the existence (or non-existence) of odd cycles is artificial in the context of Gromov hyperbolicity. However, it is an essential hypothesis in the works on direct products (see Theorem 1). Throughout the development of this work, we have verified that the existence of odd cycles is also essential in the study of hyperbolic product graphs.

2. Definitions and Background

Let ( X , d ) be a metric space, and denote by L the length associated to the distance d. A geodesic is a curve g : [ a , b ] X satisfying L ( g | [ t , s ] ) = d ( g ( t ) , g ( s ) ) = | t s | for every s , t [ a , b ] (here, g | [ t , s ] is the restriction of g to [ t , s ] ). We say that the metric space X is a geodesic metric space if for each p , q X there is a geodesic connecting them; we denote by [ p q ] any geodesic form p to q. Hence, a geodesic metric space is a connected space. When X is a graph and p , q V ( X ) , [ p , q ] denotes the edge connecting p and q if they are adjacent.
Along this paper, we consider the graphs as geodesic metric spaces. To do that, we identify any edge [ p , q ] E ( G ) with the real interval [ 0 , 1 ] ; therefore, the points in a graph are the vertices and also the points in the interior of the edges. Hence, we can define a natural distance on the points of a connected graph G by taking shortest paths in G, and so, we consider G as a metric graph. If p and q are points in different connected components of the graph, we define d ( p , q ) = .
Some authors do not consider the internal points of edges in the study. Although this approach has some advantages, we prefer to consider the internal points since these graphs are geodesic metric spaces. We use this approach since to work with geodesic metric spaces provides an interesting geometric viewpoint (for instance, Theorem 2 holds for geodesic metric spaces).
Given a geodesic metric space X and three points x 1 , x 2 , x 3 X , the geodesic triangle T = { x 1 , x 2 , x 3 } is the union of three geodesics [ x 1 x 2 ] , [ x 2 x 3 ] and [ x 3 x 1 ] . The points x 1 , x 2 , x 3 are the vertices of the triangle T. The geodesic triangle T is δ -thin if any side of T is contained in the δ -neighborhood of the union of the two other sides. We define the thin constant of the triangle T by δ ( T ) : = inf { δ 0 : T is δ thin } , and the hyperbolicity constant of the space X as δ ( X ) : = sup { δ ( T ) : T is a geodesic triangle in X } . The space X is hyperbolic if δ ( X ) < , and it is δ -hyperbolic if X is hyperbolic and the constant δ satisfies δ δ ( X ) . We say that a triangle with two identical vertices is a “bigon”. Of course, each bigon in a space (which is δ -hyperbolic) is δ -thin. If { X i } i I are the connected components of X, then we can define δ ( X ) : = sup i I δ ( X i ) , and X is hyperbolic if and only if δ ( X ) < .
We want to remark that in the classical references on hyperbolicity [14,15,77] appear many different definitions of Gromov hyperbolicity. However, the definitions are equivalent: if X is δ 1 -hyperbolic for a definition, then it is δ 2 -hyperbolic for every definition, where the constant δ 2 can be obtained from δ 1 .
We refer to the classical book [1] for definitions and background about direct product graphs.
We need bounds for the distance between points in the direct product. We use the definition given in [1].
Definition 1.
Let G 1 = ( V ( G 1 ) , E ( G 1 ) ) and G 2 = ( V ( G 2 ) , E ( G 2 ) ) be two graphs. Thedirect product G 1 × G 2 of G 1 and G 2 has V ( G 1 ) × V ( G 2 ) as vertex set, so that two distinct vertices ( u 1 , v 1 ) and ( u 2 , v 2 ) of G 1 × G 2 are adjacent if [ u 1 , u 2 ] E ( G 1 ) and [ v 1 , v 2 ] E ( G 2 ) .
If G 1 and G 2 are isomorphic, we write G 1 G 2 . It is clear that, if G 1 G 2 , then δ ( G 1 ) = δ ( G 2 ) .
It is clear that the direct product of two graphs is commutative, i.e., G 1 × G 2 G 2 × G 1 . Therefore, the conclusion of every result in this paper with some “non-symmetric” hypothesis also holds if we change the roles of G 1 and G 2 (see, e.g., Theorems 3, 4, 11, 12 and 15 and Corollary 5).
Denote by π i the projection map π i : V ( G 1 × G 2 ) V ( G i ) for i { 1 , 2 } . In fact, this projection is well defined as a map π i : G 1 × G 2 G i for i { 1 , 2 } .
We need some previous results of [1]. If u , u V ( G ) , then by a u , u -walk in G we mean a path joining u and u where repeating vertices is allowed.
Proposition 1.
([1], Proposition 5.7) Suppose ( u , v ) and ( u , v ) are vertices of the direct product G 1 × G 2 , and n is an integer for which G 1 has a u , u -walk of length n and G 2 has a v , v -walk of length n. Then, G 1 × G 2 has a walk of length n from ( u , v ) to ( u , v ) . The smallest such n (if it exists) equals d G 1 × G 2 ( ( u , v ) , ( u , v ) ) . If no such n exists, then d G 1 × G 2 ( ( u , v ) , ( u , v ) ) = .
Proposition 2.
([1], Proposition 5.8) Suppose x and y are vertices of G 1 × G 2 . Then,
d G 1 × G 2 ( x , y ) = min n N | e a c h f a c t o r G i h a s a π i ( x ) , π i ( y ) w a l k of l e n g t h n f o r i = 1 , 2 ,
where it is understood that d G 1 × G 2 ( x , y ) = if no such n exists.
Definition 2.
If G is a connected graph, the diameter of its vertices is
diam V ( G ) : = sup { d G ( u , v ) : u , v V ( G ) } ,
and the diameter of G is
diam G : = sup { d G ( x , y ) : x , y G } .
Corollary 1.
We have for every ( u , v ) , ( u , v ) V ( G 1 × G 2 )
d G 1 × G 2 ( ( u , v ) , ( u , v ) ) max d G 1 ( u , u ) , d G 2 ( v , v )
and, consequently,
diam V ( G 1 × G 2 ) max diam V ( G 1 ) , diam V ( G 2 ) .
Furthermore, if d G 1 ( u , u ) and d G 2 ( v , v ) have the same parity, then
d G 1 × G 2 ( ( u , v ) , ( u , v ) ) = max d G 1 ( u , u ) , d G 2 ( v , v )
and, consequently,
diam V ( G 1 × G 2 ) = max diam V ( G 1 ) , diam V ( G 2 ) .
By trivial graph, we mean a graph which has only a vertex.
The following result characterizes when a direct product is connected. 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.
Theorem 1.
([1], Theorem 5.9) Suppose G 1 and G 2 are connected non-trivial graphs. If at least one of G 1 or G 2 has an odd cycle, then G 1 × G 2 is connected. If both G 1 and G 2 are bipartite, then G 1 × G 2 has exactly two connected components.
Corollary 2.
([1], Corollary 5.10) A direct product of connected non-trivial graphs is connected if and only if at most one of the factors is bipartite. In fact, the product has 2 max { k , 1 } 1 connected components, where k is the number of bipartite factors.
Consider the metric spaces ( X , d X ) and ( Y , d Y ) . Given constants α 1 , β 0 , a map f : X Y is an ( α , β ) -quasi-isometric embedding if
α 1 d X ( x , y ) β d Y ( f ( x ) , f ( y ) ) α d X ( x , y ) + β ,
for x , y X . We say that f is ε -full if for each y Y there is x X with d Y ( f ( x ) , y ) ε .
We say that f is a quasi-isometry if there exist constants α , β , ε , such that f is an ε -full ( α , β ) -quasi-isometric embedding.
Two metric spaces X and Y are quasi-isometric if there exists a quasi-isometry f : X Y . One can check that to be quasi-isometric is an equivalence relation. An ( α , β ) -quasi-geodesic in X is an ( α , β ) -quasi-isometric embedding between an interval of R and X.
We need the following result ([15], p. 88).
Theorem 2
(Invariance of hyperbolicity). Let f : X Y be an ( α , β ) -quasi-isometric embedding between the geodesic metric spaces X and Y. If Y is δ Y -hyperbolic, then X is δ X -hyperbolic, where δ X is a constant which just depends on α , β , δ Y .
Besides, if f is ε-full for some ε 0 (a quasi-isometry) and X is δ X -hyperbolic, then Y is δ Y -hyperbolic, where δ Y is a constant which just depends on α , β , δ X , ε .
There are several explicit expressions for δ X = δ X ( α , β , δ Y ) , some of them very complicated. In [78] appears the best possible formula for δ X :
δ X ( α , β , δ Y ) = 8 α ( 2 α 2 ( A 1 b + A 2 δ Y ) + 4 δ Y + β ) .
for some explicit constants A 1 , A 2 .

3. Hyperbolic Direct Products

Let us start with a necessary condition for hyperbolicity.
Proposition 3.
Let G 1 and G 2 be two unbounded connected graphs. Then, G 1 × G 2 is not hyperbolic.
Proof. 
Since G 1 and G 2 are unbounded graphs, for each positive integer n there exist two geodesic paths P 1 : = [ w 1 , w 2 ] [ w 2 , w 3 ] [ w n 1 , w n ] in G 1 and P 2 : = [ v 1 , v 2 ] [ v 2 , v 3 ] [ v n 1 , v n ] in G 2 . If n is odd, then we can consider the geodesic triangle T in G 1 × G 2 (see Figure 1) defined by the following geodesics:
γ 1 : = [ ( w 1 , v 2 ) , ( w 2 , v 1 ) ] [ ( w 2 , v 1 ) , ( w 3 , v 2 ) ] [ ( w 3 , v 2 ) , ( w 4 , v 1 ) ] [ ( w n 1 , v 1 ) , ( w n , v 2 ) ] , γ 2 : = [ ( w 1 , v 2 ) , ( w 2 , v 3 ) ] [ ( w 2 , v 3 ) , ( w 1 , v 4 ) ] [ ( w 1 , v 4 ) , ( w 2 , v 5 ) ] [ ( w 1 , v n 1 ) , ( w 2 , v n ) ] , γ 3 : = [ ( w 2 , v n ) , ( w 3 , v n 1 ) ] [ ( w 3 , v n 1 ) , ( w 4 , v n 2 ) ] [ ( w 4 , v n 2 ) , ( w 5 , v n 3 ) ] [ ( w n 1 , v 3 ) , ( w n , v 2 ) ] ,
Corollary 1 gives that γ 1 , γ 2 , γ 3 are geodesics.
Let m : = n + 1 2 and consider the vertex ( w m , v m + 1 ) in γ 3 . For every vertex ( w i , v j ) in γ 1 , j { 1 , 2 } , we have d G 1 × G 2 ( ( w m , v m + 1 ) , ( w i , v j ) ) d G 2 ( v m + 1 , v j ) m + 1 2 = n 1 2 by Corollary 1. We have for every vertex ( w i , v j ) in γ 2 , i { 1 , 2 } , by Corollary 1, d G 1 × G 2 ( ( w m , v m + 1 ) , ( w i , v j ) ) d G 1 ( w m , w i ) m 2 = n 3 2 . Hence, d G 1 × G 2 ( w m , v m + 1 ) , γ 1 γ 2 n 3 2 and δ ( G 1 × G 2 ) δ ( T ) n 3 2 . Since n is arbitrarily large, G 1 × G 2 is not hyperbolic. □
Lemma 1.
Consider two connected graphs G 1 and G 2 . If f : V ( G 1 ) V ( G 2 ) is an ( α , β ) -quasi-isometric embedding, then there exists an ( α , α + β ) -quasi-isometric embedding g : G 1 G 2 with g = f on V ( G 1 ) . In addition, if f is ε-full, then g is ( ε + 1 2 ) -full.
Proof. 
For each x G 1 , let us choose a closest point v x V ( G 1 ) from x, and define g ( x ) : = f ( v x ) . Note that v x = x if x V ( G 1 ) and so g = f on V ( G 1 ) . Given x , y G 1 , we have
d G 2 ( g ( x ) , g ( y ) ) = d G 2 ( f ( v x ) , f ( v y ) ) α d G 1 ( v x , v y ) + β α d G 1 ( x , y ) + 1 + β , d G 2 ( g ( x ) , g ( y ) ) = d G 2 ( f ( v x ) , f ( v y ) ) α 1 d G 1 ( v x , v y ) β α 1 d G 1 ( x , y ) 1 β ,
and g is an ( α , α + β ) -quasi-isometric embedding, since α 1 α 1 .
In addition, if f is ε -full, then g is ( ε + 1 2 ) -full since g ( G 1 ) = f ( V ( G 1 ) ) . □
Given a graph G, let g I ( G ) denote the odd girth of G, that is, the length of the shortest odd cycle in G.
Theorem 3.
Let G 1 be a connected graph and G 2 be a non-trivial bounded connected graph with some odd cycle. Then, G 1 × G 2 is hyperbolic if and only if G 1 is hyperbolic.
Proof. 
Fix v 0 V ( G 2 ) with v 0 contained in an odd cycle C with L ( C ) = g I ( G 2 ) . Consider the map i : V ( G 1 ) V ( G 1 × G 2 ) such that i ( w ) : = ( w , v 0 ) for every w V ( G 1 ) .
By Corollary 1, for every w 1 , w 2 V ( G 1 ) , d G 1 ( w 1 , w 2 ) d G 1 × G 2 ( w 1 , v 0 ) , ( w 2 , v 0 ) . In addition, Proposition 2 gives the following.
If a geodesic joining w 1 and w 2 has even length, then
d G 1 × G 2 ( w 1 , v 0 ) , ( w 2 , v 0 ) = d G 1 ( w 1 , w 2 ) .
If a geodesic joining w 1 and w 2 has odd length, then C defines a v 0 , v 0 -walk with odd length and
d G 1 × G 2 ( w 1 , v 0 ) , ( w 2 , v 0 ) max { d G 1 ( w 1 , w 2 ) , g I ( G 2 ) } d G 1 ( w 1 , w 2 ) + g I ( G 2 ) .
Thus, i is a 1 , g I ( G 2 ) quasi-isometric embedding.
Consider any ( w , v ) V ( G 1 × G 2 ) . Then, if the geodesic joining v and v 0 has even length,
d G 1 × G 2 ( w , v ) , ( w , v 0 ) = d G 2 ( v , v 0 ) .
If a geodesic joining v and v 0 has odd length, [ v v 0 ] C defines a v , v 0 -walk with even length. Therefore,
d G 1 × G 2 ( w , v ) , ( w , v 0 ) d G 2 ( v , v 0 ) + g I ( G 2 ) .
Thus, i is diam ( V ( G 2 ) ) + g I ( G 2 ) -full.
Hence, by Lemma 1, there is a diam ( V ( G 2 ) ) + g I ( G 2 ) + 1 2 -full 1 , g I ( G 2 ) + 1 -quasi-isometry, j : G 1 G 1 × G 2 , and G 1 × G 2 is hyperbolic if and only if G 1 is hyperbolic by Theorem 2. □
Theorem 4.
Let G 1 be a connected graph without odd cycles and G 2 be a non-trivial bounded connected graph without odd cycles. Then, G 1 × G 2 is hyperbolic if and only if G 1 is hyperbolic.
Proof. 
Fix some vertex w 0 V ( G 1 ) and some edge [ v 1 , v 2 ] E ( G 2 ) .
By Theorem 1, there are exactly two components in G 1 × G 2 . Since there are no odd cycles, there is no ( w 0 , v 1 ) , ( w 0 , v 2 ) -walk in G 1 × G 2 . Thus, let us denote by ( G 1 × G 2 ) 1 the component containing the vertex ( w 0 , v 1 ) and by ( G 1 × G 2 ) 2 the component containing the vertex ( w 0 , v 2 ) .
Consider i : V ( G 1 ) V ( G 1 × G 2 ) 1 defined as i ( w ) : = ( w , v 1 ) for every w V ( G 1 ) such that every w 0 , w -walk has even length and i ( w ) : = ( w , v 2 ) for every w V ( G 1 ) such that every w 0 , w -walk has odd length.
By Proposition 2, d G 1 × G 2 i ( w 1 ) , i ( w 2 ) = d G 1 ( w 1 , w 2 ) for every w 1 , w 2 V ( G 1 ) and i is a ( 1 , 0 ) -quasi-isometric embedding.
Let ( w , v ) V ( G 1 × G 2 ) 1 . Let v j with j { 1 , 2 } such that every v , v j -walk has even length. Then, by Proposition 2, d G 1 × G 2 ( w , v ) , ( w , v j ) = d G 2 ( v , v j ) diam ( G 2 ) . Therefore, i is diam ( G 2 ) -full.
Hence, by Lemma 1, there is a diam ( G 2 ) + 1 2 -full 1 , 1 -quasi-isometry, j : G 1 ( G 1 × G 2 ) 1 , and ( G 1 × G 2 ) 1 is hyperbolic if and only if G 1 is hyperbolic by Theorem 2.
The same argument proves that ( G 1 × G 2 ) 2 is hyperbolic. □
Denote by P 2 the path graph with two vertices and an edge.
Lemma 2.
Let G 1 be a connected graph with some odd cycle and G 2 a non-trivial bounded graph without odd cycles. Then, G 1 × G 2 and G 1 × P 2 are quasi-isometric and δ ( G 1 × P 2 ) δ ( G 1 × G 2 ) .
Proof. 
By Theorem 1, we know that G 1 × G 2 and G 1 × P 2 are connected graphs.
Denote by v 1 and v 2 the vertices of P 2 and fix [ w 1 , w 2 ] E ( G 2 ) . The map f : V ( G 1 × P 2 ) V ( G 1 × [ w 1 , w 2 ] ) defined as f ( u , v j ) : = ( u , w j ) for every u V ( G 1 ) and j = 1 , 2 , is an isomorphism of graphs; hence, it suffices to prove that G 1 × G 2 and G 1 × [ w 1 , w 2 ] are quasi-isometric.
Consider the inclusion map i : V ( G 1 × [ w 1 , w 2 ] ) V ( G 1 × G 2 ) . Since G 1 × [ w 1 , w 2 ] is a subgraph of G 1 × G 2 , we have d G 1 × G 2 ( x , y ) d G 1 × [ w 1 , w 2 ] ( x , y ) for every x , y V ( G 1 × [ w 1 , w 2 ] ) .
Since G 2 is a graph without odd cycles, every w 1 , w 2 -walk has odd length and every w j , w j -walk has even length for j = 1 , 2 . Thus, Proposition 2 gives, for every x = ( u , w 1 ) , y = ( v , w 2 ) V ( G 1 × [ w 1 , w 2 ] ) ,
d G 1 × [ w 1 , w 2 ] ( x , y ) = d G 1 × G 2 ( x , y ) = min L ( g ) | g is a u , v - walk of odd length .
Furthermore, for every x = ( u , w j ) , y = ( v , w j ) V ( G 1 × [ w 1 , w 2 ] ) and j = 1 , 2 ,
d G 1 × [ w 1 , w 2 ] ( x , y ) = d G 1 × G 2 ( x , y ) = min L ( g ) | g is a u , v - walk of even length .
Hence, d G 1 × [ w 1 , w 2 ] ( x , y ) = d G 1 × G 2 ( x , y ) for every x , y V ( G 1 × [ w 1 , w 2 ] ) , and the inclusion map i is an ( 1 , 0 ) -quasi-isometric embedding. Therefore, δ ( G 1 × P 2 ) = δ ( G 1 × [ w 1 , w 2 ] ) δ ( G 1 × G 2 ) .
Since G 2 is a graph without odd cycles, given any w V ( G 2 ) , we have either that every w , w 1 -walk has even length and every w , w 2 -walk has odd length or that every w , w 2 -walk has even length and every w , w 1 -walk has odd length. In addition, since G 1 is connected, for each u V ( G 1 ) there is some u V ( G 1 ) such that [ u , u ] E ( G 1 ) . Therefore, by Proposition 2, for every ( u , w ) V ( G 1 × G 2 ) , if min d G 2 ( w , w 1 ) , d G 2 ( w , w 2 ) is even, then
d G 1 × G 2 ( u , w ) , V ( G 1 × [ w 1 , w 2 ] ) = d G 1 × G 2 ( u , w ) , V ( u × [ w 1 , w 2 ] ) = min d G 2 ( w , w 1 ) , d G 2 ( w , w 2 ) ,
and if min d G 2 ( w , w 1 ) , d G 2 ( w , w 2 ) is odd, then
d G 1 × G 2 ( u , w ) , V ( G 1 × [ w 1 , w 2 ] ) = d G 1 × G 2 ( u , w ) , V ( u × [ w 1 , w 2 ] ) = min d G 2 ( w , w 1 ) , d G 2 ( w , w 2 ) .
In both cases,
d G 1 × G 2 ( u , w ) , V ( G 1 × [ w 1 , w 2 ] ) diam V ( G 2 ) ,
and i is diam V ( G 2 ) -full. By Lemma 1, there exists a diam V ( G 2 ) + 1 2 -full ( 1 , 1 ) -quasi-isometry g : G 1 × [ w 1 , w 2 ] G 1 × G 2 . □
A subgraph Γ of G is said isometric if d Γ ( x , y ) = d G ( x , y ) for any x , y Γ . One can check that Γ is isometric if and only if d Γ ( u , v ) = d G ( u , v ) for any u , v V ( Γ ) .
Lemma 3.
([47], Lemma 5) If Γ is an isometric subgraph of G, then δ ( Γ ) δ ( G ) .
A u , v -walk g in G is a shortcut of a cycle C if g C = { u , v } and L ( g ) < d C ( u , v ) where d C denotes the length metric on C.
A cycle C is a reduction of the cycle C if both have odd length and C is the union of a subarc η of C and a shortcut of C joining the endpoints of η . Note that L ( C ) L ( C ) 2 . We say that a cycle is minimal if it has odd length and it does not have a reduction.
Lemma 4.
If C is a minimal cycle of G, then L ( C ) 4 δ ( G ) .
Proof. 
We prove first that C is an isometric subgraph of G. Assume that C is not an isometric subgraph. Thus, there exists a shortcut g of C with endpoints u , v . There are two subarcs η 1 , η 2 of C joining u and v; since C has odd length, we can assume that η 1 has even length and η 2 has odd length. If g has even length, then C : = g η 2 is a reduction of C. If g has odd length, then C : = g η 1 is a reduction of C. Hence, C is not minimal, a contradiction, and so C is an isometric subgraph of G.
It is easy to show that any isometric cycle C has length 4 δ ( C ) . This fact and Lemma 3 give L ( C ) = 4 δ ( C ) 4 δ ( G ) . □
Given any w 0 , w k -walk g = [ w 0 , w 1 ] [ w 1 , w 2 ] [ w k 1 , w k ] in G 1 and P 2 = [ v 1 , v 2 ] , if L ( g ) is either odd or even, then we define the ( w 0 , v 1 ) , ( w k , v i ) -walk for i 1 , 2 ,
Γ 1 g : = [ ( w 0 , v 1 ) , ( w 1 , v 2 ) ] [ ( w 1 , v 2 ) , ( w 2 , v 1 ) ] [ ( w 2 , v 1 ) , ( w 3 , v 2 ) ] [ ( w k 1 , v 1 ) , ( w k , v 2 ) ] , Γ 1 g : = [ ( w 0 , v 1 ) , ( w 1 , v 2 ) ] [ ( w 1 , v 2 ) , ( w 2 , v 1 ) ] [ ( w 2 , v 1 ) , ( w 3 , v 2 ) ] [ ( w k 1 , v 2 ) , ( w k , v 1 ) ] ,
respectively.
Remark 1.
By Proposition 2, if g is a geodesic path in G 1 , then Γ 1 g is a geodesic path in G 1 × P 2 .
Let us define the map R : V ( G 1 × P 2 ) V ( G 1 × P 2 ) as R ( w , v 1 ) = ( w , v 2 ) and R ( w , v 2 ) = ( w , v 1 ) for every w V ( G 1 ) , and the path Γ 2 g as Γ 2 g = R ( Γ 1 g ) .
Let us define the map ( Γ 1 g ) : g Γ 1 g which is an isometry on the edges and such that ( Γ 1 g ) ( w j ) = ( w j , v 1 ) if j is even and ( Γ 1 g ) ( w j ) = ( w j , v 2 ) if j is odd. In addition, let ( Γ 2 g ) : g Γ 2 g be the map defined by ( Γ 2 g ) : = R ( Γ 1 g ) .
Given a graph G, denote by C the set of minimal cycles of G.
Lemma 5.
Let G 1 be a connected graph with some odd cycle and P 2 = [ v 1 , v 2 ] . Consider a geodesic g = [ w 0 w k ] = [ w 0 , w 1 ] [ w 1 , w 2 ] [ w k 1 , w k ] in G 1 . Let us define w 0 : = ( Γ 1 g ) ( w 0 ) = ( w 0 , v 1 ) and w k : = ( Γ 2 g ) ( w k ) , i.e., w k : = ( w k , v 1 ) or w k : = ( w k , v 2 ) if k is odd or even, respectively. Then, d G 1 × P 2 ( w 0 , w k ) > d G 1 w j , C ( G 1 ) for every 0 j k .
Proof. 
Fix 0 j k . Define
P : = σ | σ is a w 0 , w k - walk such that L ( σ ) has a parity different from that of k .
Proposition 2 gives
d G 1 × P 2 ( w 0 , w k ) = min L ( σ ) | σ P .
Choose σ 0 P such that L ( σ 0 ) = d G 1 × P 2 ( w 0 , w k ) . Since L ( g ) + L ( σ 0 ) is odd, we have L ( g ) + L ( σ 0 ) = 2 t + 1 for some positive integer t. Thus, d G 1 × P 2 ( w 0 , w k ) = L ( σ 0 ) > 1 2 ( 2 t + 1 ) .
If g σ 0 is a cycle, then let us define C 0 : = g σ 0 . Thus, L ( C 0 ) = 2 t + 1 and d G 1 w j , C 0 = 0 for every 0 j k . Otherwise, we may assume that g σ 0 = [ w 0 w i 1 ] [ w i 2 w k ] for some 0 i 1 < i 2 k . If σ 1 = σ 0 \ g , then let us define C 0 : = [ w i 1 w i 2 ] σ 1 (where [ w i 1 w i 2 ] g ). Hence, C 0 is a cycle, L ( C 0 ) 2 t 1 and d G 1 w j , C 0 < 1 2 ( 2 t + 1 ) .
If C 0 is not minimal, then consider a reduction C 1 of C 0 . Let us repeat the process until we obtain a minimal cycle C s . Note that L ( C 1 ) L ( C 0 ) 2 and for every point p 1 C 0 , d G 1 p 1 , C 1 < 1 2 L ( C 0 ) . Now, repeating the argument, for every 1 < i s , L ( C i ) L ( C i 1 ) 2 and for every point p i C i 1 , d G 1 p i , C i < 1 2 L ( C i 1 ) . Therefore,
d G 1 w j , C ( G 1 ) d G 1 w j , C s d G 1 w j , C 0 + 1 2 L ( C 0 ) + 1 2 L ( C 1 ) + + 1 2 L ( C s ) < 1 2 ( 2 t + 1 ) + 1 2 ( 2 t 1 ) + + 5 2 + 3 2 .
Hence,
d G 1 w j , C ( G 1 ) < 1 2 i = 1 t ( 2 i + 1 ) = 1 2 t 2 + t < 1 2 ( 2 t + 1 ) 2 < d G 1 × P 2 ( w 0 , w k ) 2 .
Corollary 3.
Let G 1 be a hyperbolic connected graph with some odd cycle and P 2 = [ v 1 , v 2 ] . Consider a geodesic g = [ w 0 w k ] = [ w 0 , w 1 ] [ w 1 , w 2 ] [ w k 1 , w k ] in G 1 . Let us define w 0 : = ( Γ 1 g ) ( w 0 ) = ( w 0 , v 1 ) and w k : = ( Γ 2 g ) ( w k ) . Then, we have for every 0 j k ,
1 2 k + d G 1 w j , C ( G 1 ) d G 1 × P 2 ( w 0 , w k ) k + 2 d G 1 w j , C ( G 1 ) + 4 δ ( G 1 ) .
Proof. 
Corollary 1 and Lemma 5 give d G 1 × P 2 ( w 0 , w k ) k and d G 1 × P 2 ( w 0 , w k ) d G 1 w j , C ( G 1 ) , and these inequalities provide the lower bound of d G 1 × P 2 ( w 0 , w k ) .
Consider a geodesic γ joining w j and C C ( G 1 ) with L ( γ ) = d G 1 ( w j , C ) = d G 1 w j , C ( G 1 ) and the w 0 , w k -walk
g : = [ w 0 w j ] γ C γ [ w j w k ] .
One can check that Γ 1 g is a w 0 , w k -walk in G 1 × P 2 , and so Lemma 4 gives
d G 1 × P 2 ( w 0 , w k ) L ( Γ 1 g ) = L ( g ) = k + 2 d G 1 w j , C ( G 1 ) + L ( C ) k + 2 d G 1 w j , C ( G 1 ) + 4 δ ( G 1 ) .
If [ v 1 , v 2 ] is an edge of G, then the point x [ v 1 , v 2 ] with d G ( x , v 1 ) = d G ( x , v 2 ) = 1 / 2 is the midpoint of the edge [ v 1 , v 2 ] . Denote by J ( G ) the set of vertices and midpoints of edges in G. Consider the set T 1 ( G ) of geodesic triangles T in G which are cycles and such that the vertices of T are in J ( G ) . We denote by δ 1 ( G ) the infimum of the constants μ such that any triangle in T 1 ( G ) is μ -thin.
The following three results are used throughout the paper.
Theorem 5.
([40], Theorem 2.5) For every connected graph G, we have δ 1 ( G ) = δ ( G ) .
Theorem 6.
([40], Theorem 2.6) Let G be any connected graph. Then, δ ( G ) is always a multiple of 1 / 4 .
Theorem 7.
([40], Theorem 2.7) For any hyperbolic connected graph G, there exists a geodesic triangle T T 1 ( G ) such that δ ( T ) = δ ( G ) .
Consider the set T v ( G ) of geodesic triangles T in G that are cycles and such that the three vertices of the triangle T are also vertices of G. δ v ( G ) denotes the infimum of the constants μ such that every triangle in T v ( G ) is μ -thin.
Theorem 8.
For every connected graph G, we have δ v ( G ) δ ( G ) 4 δ v ( G ) + 1 / 2 . Hence, G is hyperbolic if and only if δ v ( G ) < . Furthermore, if G is hyperbolic, then there are a geodesic triangle T = { a , b , c } T v ( G ) and q [ a b ] J ( G ) such that d ( p , [ a c ] [ c b ] ) = δ ( T ) = δ v ( G ) . In addition, δ v ( G ) is an integer multiple of 1 / 2 .
Proof. 
The inequality δ v ( G ) δ ( G ) is direct.
Consider the set T v ( G ) of geodesic triangles T in G such that the three vertices of the triangle T belong to V ( G ) , and denote by δ v ( G ) the infimum of the constants μ such that every triangle in T v ( G ) is μ -thin. The argument in the proof of (ref. [79], Lemma 2.1) gives that δ v ( G ) = δ v ( G ) .
Let us prove now δ ( G ) 4 δ v ( G ) + 1 / 2 . Let us assume that G is hyperbolic. If δ v ( G ) = , then the inequality is trivial. Thus, it suffices to consider the case δ v ( G ) < . By Theorem 7, there is a triangle T = { a , b , c } that is a cycle with a , b , c J ( G ) and q [ a b ] such that d ( q , [ a c ] [ c b ] ) = δ ( T ) = δ ( G ) . Assume that a , b , c J ( G ) \ V ( G ) (otherwise, the argument is simpler). Let a 1 , a 2 , b 1 , b 2 , c 1 , c 2 T V ( G ) such that a [ a 1 , a 2 ] , b [ b 1 , b 2 ] , c [ c 1 , c 2 ] and a 2 , b 1 [ a b ] , c 2 , d 1 [ c d ] , d 2 , a 1 [ a c ] . Since H : = { a 2 , b 1 , b 2 , c 1 , c 2 , a 1 } is a geodesic hexagon with vertices in V ( G ) , it is 4 δ v ( G ) -thin and every point w [ b 1 , b 2 ] [ b 2 c 1 ] [ c 1 , c 2 ] [ c 2 a 1 ] [ a 1 , a 2 ] verifies d ( w , [ a c ] [ c b ] ) 1 / 2 , we have
δ ( G ) = d ( q , [ a c ] [ c b ] ) d ( q , [ b 1 , b 2 ] [ b 2 c 1 ] [ c 1 , c 2 ] [ c 2 a 1 ] [ a 1 , a 2 ] ) + 1 / 2 4 δ v ( G ) + 1 / 2 = 4 δ v ( G ) + 1 / 2 .
Assume that G is not hyperbolic. Therefore, for each M > 0 there is a triangle T = { a , b , c } which is a cycle with a , b , c J ( G ) and q [ a b ] with d ( q , [ a c ] [ c b ] ) M . The previous argument gives M 4 δ v ( G ) + 1 / 2 and, since M is arbitrary, we conclude δ v ( G ) = = δ ( G ) .
Finally, consider any geodesic triangle T = { a , b , c } in T v ( G ) . Since d ( q , [ a c ] [ c b ] ) = d ( q , ( [ a c ] [ c b ] ) V ( G ) ) , d ( q , [ a c ] [ c b ] ) attains its maximum value when q J ( G ) . Hence, δ ( T ) is a multiple of 1 / 2 for any triangle T T v ( G ) . Since the set of non-negative numbers that are multiple of 1 / 2 is a discrete set, δ ( G ) is an integer multiple of 1 / 2 if G is hyperbolic, and there is a triangle T = { a , b , c } T v ( G ) and q [ a b ] J ( G ) with d ( q , [ a c ] [ c b ] ) = δ ( T ) = δ v ( G ) . This finishes the proof. □
Theorem 9.
If G 1 is a non-hyperbolic connected graph, then G 1 × P 2 is not hyperbolic.
Proof. 
Since G 1 is not hyperbolic, by Theorem 8, given any R > 0 there exists a triangle T = { x , y , z } wich is a cycle, with x , y , z V ( G 1 ) and such that T is not R-thin. Therefore, there exists some point m T , let us assume that m [ x y ] , such that d G 1 ( m , [ y z ] [ z x ] ) > R .
Seeking for a contradiction let us assume that G 1 × P 2 is δ -hyperbolic.
Suppose that for some R > δ , there is a geodesic triangle T = { x , y , z } that is an even cycle in G 1 , with x , y , z V ( G 1 ) and such that T is not R-thin. Consider the (closed) path Λ = [ x y ] [ y z ] [ z x ] . Then, since T has even length, the path Γ 1 Λ defines a cycle in G 1 × P 2 . Let γ 1 , γ 2 , γ 3 be the paths in Γ 1 Λ corresponding to [ x y ] , [ y z ] , [ z x ] , respectively. By Corollary 1, the curves γ 1 , γ 2 and γ 3 are geodesics, and d G 1 × P 2 ( Γ 1 Λ ) ( m ) , γ 2 γ 3 > δ , leading to contradiction.
Suppose that, for every R > 0 , there is a geodesic triangle T = { x , y , z } which is an odd cycle, with x , y , z V ( G 1 ) and such that T is not R-thin.
Let T 1 = { x , y , z } be a geodesic triangle as above and let us assume that diam ( T 1 ) = D > 8 δ .
Let T 2 = { x , y , z } be another triangle as above such that T 2 is not 3 ( D + 8 δ ) -thin, this is, there is a point m in one of the sides, let us call it σ , of T 2 such that d G 1 ( m , T 2 \ σ ) > 3 ( D + 8 δ ) .
Let g = [ w 0 w k ] with w 0 T 1 and w k T 2 be a shortest geodesic in G 1 joining T 1 and T 2 (if T 1 and T 2 intersect, just assume that g is a single vertex, w 0 = w k , in the intersection). See Figure 2.
Let us assume that w 0 [ x z ] and w k [ x z ] . Then, let us consider the closed path C in G 1 given by the union of the geodesics in T 1 , g, the geodesics in T 2 and the inverse of g from w k to w 0 , this is,
C : = [ w 0 x ] [ x y ] [ y z ] [ z w 0 ] [ w 0 w k ] [ w k x ] [ x y ] [ y z ] [ z w k ] [ w k w 0 ] .
Since T 1 , T 2 are odd cycles, C is an even closed cycle. Therefore, Γ 1 C defines a cycle in G 1 × P 2 . Moreover, by Remark 1, Γ 1 C is a geodesic decagon in G 1 × P 2 with sides γ 1 = ( Γ 1 C ) ( [ w 0 x ] ) , γ 2 = ( Γ 1 C ) ( [ x y ] ) , γ 3 = ( Γ 1 C ) ( [ y z ] ) , γ 4 = ( Γ 1 C ) ( [ z w 0 ] ) , γ 5 = ( Γ 1 C ) ( [ w 0 w k ] ) , γ 6 = ( Γ 1 C ) ( [ w k x ] ) , γ 7 = ( Γ 1 C ) ( [ x y ] ) , γ 8 = ( Γ 1 C ) ( [ y z ] ) , γ 9 = ( Γ 1 C ) ( [ z w k ] ) and γ 10 = ( Γ 1 C ) ( [ w k w 0 ] ) .
Since we are assuming that G 1 × P 2 is δ -hyperbolic, then for every 1 i 10 and every point p γ i , d G 1 × P 2 ( p , C \ γ i ) 8 δ .
Let p : = ( Γ 1 C ) ( m ) .
Case 1. Suppose that d G 1 ( m , T 1 g ) > 8 δ . See Figure 2.
By assumption, d G 1 ( m , T 2 \ σ ) > 8 δ . If σ = [ x y ] (resp. σ = [ y z ] ), then p γ 7 (resp. p γ 8 ) and, by Corollary 1, d G 1 × P 2 ( p , C \ γ 7 ) > 8 δ (resp. d G 1 × P 2 ( p , C \ γ 8 ) > 8 δ ) leading to contradiction. If σ = [ x z ] , since [ x z ] = [ x w k ] [ w k z ] , let us assume m [ x w k ] . Then, since d G 1 ( m , w k ) > 8 δ , it follows that d G 1 ( m , [ w k z ] ) > 8 δ . Thus, p γ 6 and, by Corollary 1, d G 1 × P 2 ( p , C \ γ 6 ) > 8 δ leading to contradiction.
Case 2. Suppose that d G 1 ( m , T 1 g ) 8 δ and L ( g ) 8 δ . See the left side of Figure 3. Then, for every point q in T 1 g , d G 1 ( m , q ) 8 δ + D + 8 δ . In particular, d G 1 ( m , w k ) 8 δ + D + 8 δ . Therefore, m [ x z ] and let us assume that m [ x w k ] . Since d G 1 ( m , x ) d G 1 ( m , [ x y ] [ y z ] ) > 3 ( D + 8 δ ) , there is a point m [ x m ] [ x w k ] such that d G 1 ( m , m ) = 2 ( D + 8 δ ) . Then, d G 1 ( m , T 1 g ) 2 ( D + 8 δ ) D 8 δ 8 δ = D > 8 δ . In addition, it is trivial to check that d G 1 ( m , [ x y ] [ y z ] ) > 3 ( D + 8 δ ) 2 ( D + 8 δ ) > 8 δ and since [ x z ] is a geodesic, d G 1 ( m , [ z w k ] ) > 8 δ . Thus, if p : = ( Γ 1 C ) ( m ) , then p γ 6 and, by Corollary 1, d G 1 × P 2 ( p , C \ γ 6 ) > 8 δ leading to contradiction.
Case 3. Suppose that d G 1 ( m , T 1 g ) 8 δ and L ( g ) > 8 δ . See the right side of Figure 3. Since g is a shortest geodesic in G 1 joining T 1 and T 2 , this implies that d G 1 ( T 1 , T 2 ) > 8 δ and d G 1 ( m , [ w 0 w k ] ) 8 δ . Moreover, d G 1 ( m , w k ) 16 δ . Otherwise, there is a point q [ w 0 w k ] such that d G 1 ( m , q ) 8 δ and d G 1 ( q , w k ) > 8 δ which means that d G 1 ( q , w 0 ) < d G 1 ( w 0 , w k ) 8 δ and d G 1 ( m , w 0 ) < d G 1 ( w 0 , w k ) leading to contradiction.
Since d G 1 ( m , w k ) 16 δ , m [ x z ] . Let us assume that m [ x w k ] . Since d G 1 ( m , [ x y ] [ y z ] ) > 3 ( D + 8 δ ) , there is a point m [ x m ] [ x w k ] such that d G 1 ( m , m ) = 2 ( D + 8 δ ) . Let us see that d G 1 ( m , [ w 0 w k ] ) > 8 δ . Suppose there is some q [ w 0 w k ] such that d G 1 ( m , q ) 8 δ . Since m T 2 and g is a shortest geodesic joining T 1 and T 2 , d G 1 ( q , w k ) 8 δ . However, 32 δ < 2 ( D + 8 δ ) = d G 1 ( m , m ) d G 1 ( m , q ) + d G 1 ( q , w k ) + d G 1 ( w k , m ) 8 δ + 8 δ + 16 δ which is a contradiction. Hence, d G 1 ( m , [ w 0 w k ] ) > 8 δ . In addition, it is trivial to check that d G 1 ( m , [ x y ] [ y z ] ) > 3 ( D + 8 δ ) 2 ( D + 8 δ ) > 8 δ and since [ x z ] is a geodesic, d G 1 ( m , [ z w k ] ) > 8 δ . Thus, if p : = ( Γ 1 C ) ( m ) , then p γ 6 and, by Corollary 1, d G 1 × P 2 ( p , C \ γ 6 ) > 8 δ leading to contradiction. □
Proposition 3, Lemma 2 and Theorems 3, 4 and 9 have the following consequence.
Corollary 4.
If G 1 is a non-hyperbolic connected graph and G 2 is some non-trivial connected graph, then G 1 × G 2 is not hyperbolic.
Proposition 3 and Corollary 4 provide a necessary condition for the hyperbolicity of G 1 × G 2 .
Theorem 10.
Let G 1 , G 2 be non-trivial connected graphs. If G 1 × G 2 is hyperbolic, then one factor graph is hyperbolic and the other one is bounded.
Theorems 3 and 4 show that this necessary condition is also sufficient if either G 2 has some odd cycle or G 1 and G 2 do not have odd cycles (when G 1 is a hyperbolic graph and G 2 is a bounded graph). We deal now with the other case, when G 1 has some odd cycle and G 2 does not have odd cycles.
Theorem 11.
Let G 1 be a connected graph with some odd cycle and G 2 a non-trivial bounded connected graph without odd cycles. Assume that G 1 satisfies the following property: for each M > 0 there exist a geodesic g joining two minimal cycles of G 1 and a vertex u g V ( G 1 ) with d G 1 u , C ( G 1 ) M . Then, G 1 × G 2 is not hyperbolic.
Proof. 
If G 1 is not hyperbolic, then Corollary 4 gives that G 1 × G 2 is not hyperbolic. Assume now that G 1 is hyperbolic. By Theorem 2 and Lemma 2, we can assume that G 2 = P 2 and V ( P 2 ) = { v 1 , v 2 } .
Fix M > 0 and choose a geodesic g = [ w 0 w k ] = [ w 0 , w 1 ] [ w 1 , w 2 ] [ w k 1 , w k ] joining two minimal cycles in G 1 and 0 < r < k with d G 1 w r , C ( G 1 ) M .
Define the paths g 1 and g 2 in G 1 × P 2 as g 1 : = Γ 1 g and g 2 : = Γ 2 g . Since L ( g 1 ) = L ( g 2 ) = L ( g ) = d G 1 ( w 0 , w k ) , we have
d G 1 × P 2 g 1 ( w 0 ) , g 1 ( w k ) L ( g 1 ) = d G 1 ( w 0 , w k ) , d G 1 × P 2 g 2 ( w 0 ) , g 2 ( w k ) L ( g 2 ) = d G 1 ( w 0 , w k ) .
Corollary 1 gives that
d G 1 × P 2 g 1 ( w 0 ) , g 1 ( w k ) d G 1 ( w 0 , w k ) , d G 1 × P 2 g 2 ( w 0 ) , g 2 ( w k ) d G 1 ( w 0 , w k ) .
Hence, g 1 and g 2 are geodesics in G 1 × P 2 . Choose geodesics g 3 = [ g 1 ( w 0 ) g 2 ( w 0 ) ] and g 4 = [ g 1 ( w k ) g 2 ( w k ) ] in G 1 × P 2 . Since d P 2 ( v 1 , v 2 ) = 1 is odd, Proposition 2 gives
d G 1 × P 2 g 1 ( w 0 ) , g 2 ( w 0 ) = min L ( σ ) | σ is a w 0 , w 0 - walk = min L ( σ ) | σ cycle of odd length containing w 0 .
Since w 0 belongs to a minimal cycle, L ( g 3 ) 4 δ ( G 1 ) by Lemma 4. In a similar way, we obtain L ( g 4 ) 4 δ ( G 1 ) .
Consider the geodesic quadrilateral Q : = { g 1 , g 2 , g 3 , g 4 } in G 1 × P 2 . Thus, d G 1 × P 2 g 1 ( w r ) , g 2 g 3 g 4 2 δ ( G 1 × P 2 ) . Since max L ( g 3 ) , L ( g 4 ) 4 δ ( G 1 ) , we deduce d G 1 × P 2 g 1 ( w r ) , g 2 2 δ ( G 1 × P 2 ) + 4 δ ( G 1 ) .
Let 0 j k with d G 1 × P 2 g 1 ( w r ) , g 2 = d G 1 × P 2 g 1 ( w r ) , g 2 ( w j ) . Let us define w r : = g 1 ( w r ) and w j : = g 2 ( w j ) . Thus, Lemma 5 gives
M d G 1 w r , C ( G 1 ) d G 1 × P 2 ( w r , w j ) = d G 1 × P 2 ( w r , g 2 ) 2 δ ( G 1 × P 2 ) + 4 δ ( G 1 ) ,
and since M is arbitrarily large, we deduce that G 1 × P 2 is not hyperbolic. □
Lemma 6.
Let G 1 be a hyperbolic connected graph and suppose there is some constant K > 0 such that for every vertex w G 1 , d G 1 ( w , C ( G 1 ) ) K . Then, G 1 × P 2 is hyperbolic.
Proof. 
Denote by v 1 and v 2 the vertices of P 2 . Let i : V ( G 1 ) V ( G 1 × P 2 ) defined as i ( w ) : = ( w , v 1 ) for every w G 1 .
For every x , y V ( G 1 ) , by Corollary 1, d G 1 ( x , y ) d G 1 × P 2 ( i ( x ) , i ( y ) ) . By Corollary 3,
d G 1 × P 2 ( i ( x ) , i ( y ) ) d G 1 ( x , y ) + 2 d G 1 x , C ( G 1 ) + 4 δ ( G 1 ) d G 1 ( x , y ) + 2 K + 4 δ ( G 1 ) .
Therefore, i : V ( G 1 ) V ( G 1 × P 2 ) is a 1 , 2 K + 4 δ ( G 1 ) -quasi-isometric embedding.
Notice that for every ( w , v 1 ) V ( G 1 × P 2 ) , ( w , v 1 ) = i ( w ) . In addition, for any ( w , v 2 ) V ( G 1 × P 2 ) , since G 1 is connected, there is some edge [ w , w ] E ( G 1 ) and we have [ ( w , v 2 ) , ( w , v 1 ) ] E ( G 1 × P 2 ) . Therefore, i : V ( G 1 ) V ( G 1 × P 2 ) is 1-full.
Thus, by Lemma 1, G 1 and G 1 × P 2 are quasi-isometric and, by Theorem 2, G 1 × P 2 is hyperbolic. □
Theorem 3 and Lemmas 2 and 6 give the following result.
Theorem 12.
Let G 1 be a hyperbolic connected graph and G 2 some non-trivial bounded connected graph. If there is some constant K > 0 such that for every vertex w G 1 , d G 1 ( w , C ( G 1 ) ) K , then G 1 × G 2 is hyperbolic.
We finish this section with a characterization of the hyperbolicity of G 1 × G 2 , under an additional hypothesis. We present first some lemmas.
Let J be a finite or infinite index set. Now, given a graph G 1 , we define some graphs related to G 1 which will be useful in the following results. Let B j : = B G 1 ( w j , K j ) with w j V ( G 1 ) and K j Z + , for any j J , such that sup j K j = K < , B ¯ j 1 B ¯ j 2 = if j 1 j 2 , and every odd cycle C in G 1 satisfies C B j for some j J . Denote by G 1 the subgraph of G 1 induced by V ( G 1 ) \ ( j B j ) . Let N j : = B j = { w V ( G 1 ) : d G 1 ( w , w j ) = K j } . Denote by G 1 the graph with V ( G 1 ) = V ( G 1 ) ( j { w j } ) , where w j are additional vertices, and E ( G 1 ) = E ( G 1 ) ( j { [ w , w j ] : w N j } ) . We have G 1 = G 1 G 1 .
Lemma 7.
Let G 1 be a connected graph as above. Then, there is a quasi-isometry g : G 1 G 1 such that g ( w j ) = w j for every j J .
Proof. 
Let f : V ( G 1 ) V ( G 1 ) defined as f ( u ) = u for every u V ( G 1 ) , and f ( u ) = w i for every u V ( B i ) . It is clear that f : V ( G 1 ) V ( G 1 ) is 0-full.
Now, we focus on proving that f : V ( G 1 ) V ( G 1 ) is a ( K , 2 K ) -quasi-isometric embedding. For every u , v V ( G 1 ) , it is clear that d G 1 ( f ( u ) , f ( v ) ) d G 1 ( u , v ) .
Let us prove the other inequality. Fix u , v V ( G 1 ) and consider an oriented geodesic γ in G 1 from f ( u ) to f ( v ) .
Assume that u , v V ( G 1 ) . If L ( γ ) = d G 1 ( u , v ) , then d G 1 ( u , v ) = d G 1 ( f ( u ) , f ( v ) ) . If L ( γ ) < d G 1 ( u , v ) , then γ meets some w j . Since γ is a compact set, it intersects only a finite number of w j ’s, which we denote by w j 1 , w j r . Since γ is an oriented curve from f ( u ) to f ( v ) , we can assume that γ meets w j 1 , w j r in this order.
Let us define the following vertices in γ
w i 1 = [ f ( u ) w j i ] N j i , w i 2 = [ w j i f ( v ) ] N j i ,
for every 1 i r . Note that [ w i 2 w i + 1 1 ] G 1 for every 1 i < r (it is possible to have w i 2 = w i + 1 1 ).
Since d G 1 ( w i 1 , w i 2 ) = 2 and d G 1 ( w i 1 , w i 2 ) 2 K , we have d G 1 ( w i 1 , w i 2 ) 1 K d G 1 ( w i 1 , w i 2 ) for every 1 i r . Thus,
d G 1 ( f ( u ) , f ( v ) ) = d G 1 ( f ( u ) , w 1 1 ) + i = 1 r d G 1 ( w i 1 , w i 2 ) + i = 1 r 1 d G 1 ( w i 2 , w i + 1 1 ) + d G 1 ( w r 2 , f ( v ) ) d G 1 ( u , w 1 1 ) + 1 K i = 1 r d G 1 ( w i 1 , w i 2 ) + i = 1 r 1 d G 1 ( w i 2 , w i + 1 1 ) + d G 1 ( w r 2 , v ) 1 K d G 1 ( u , w 1 1 ) + i = 1 r d G 1 ( w i 1 , w i 2 ) + i = 1 r 1 d G 1 ( w i 2 , w i + 1 1 ) + d G 1 ( w r 2 , v ) 1 K d G 1 ( u , v ) .
Assume that f ( u ) = f ( v ) . Therefore, there exists j with u , v B j and
d G 1 ( f ( u ) , f ( v ) ) = 0 > d G 1 ( u , v ) 2 K .
Assume now that u and/or v does not belong to V ( G 1 ) and f ( u ) f ( v ) . Let u 0 , v 0 be the closest vertices in V ( G 1 ) γ to f ( u ) , f ( v ) , respectively (it is possible to have u 0 = f ( u ) or v 0 = f ( v ) ). Since u 0 , v 0 V ( G 1 ) , u 0 = f ( u 0 ) , v 0 = f ( v 0 ) , we have d G 1 ( u , u 0 ) < 2 K and d G 1 ( v , v 0 ) < 2 K . Hence,
d G 1 ( f ( u ) , f ( v ) ) = d G 1 ( f ( u ) , u 0 ) + d G 1 ( u 0 , v 0 ) + d G 1 ( v 0 , f ( v ) ) d G 1 ( f ( u 0 ) , f ( v 0 ) ) 1 K d G 1 ( u 0 , v 0 ) 1 K d G 1 ( u , v ) d G 1 ( u , u 0 ) d G 1 ( v , v 0 ) > 1 K d G 1 ( u , v ) 4 .
If K 2 , then d G 1 ( f ( u ) , f ( v ) ) > 1 K d G 1 ( u , v ) 2 K . If K = 1 , then d G 1 ( u , u 0 ) 1 , d G 1 ( v , v 0 ) 1 , and d G 1 ( f ( u ) , f ( v ) ) d G 1 ( u , v ) 2 .
Finally, we conclude that f : V ( G 1 ) V ( G 1 ) is a ( K , 2 K ) -quasi-isometric embedding. Thus, Lemma 1 provides a quasi-isometry g : G 1 G 1 with the required property. □
Definition 3.
Given a connected graph G 1 and some index set J, let B J = { B j } j J be a family of balls where B j : = B G 1 ( w j , K j ) with w j V ( G 1 ) , K j Z + for any j J , sup j K j = K < and B ¯ j 1 B ¯ j 2 = if j 1 j 2 . Suppose that every odd cycle C in G 1 satisfies that C B j for some j J . If there is some constant M > 0 such that for every j J , there is an odd cycle C j such that C j B j with L ( C j ) < M , then we say that B J is M-regular.
Remark 2.
If J is finite, then there exists M > 0 such that { B j } j J is M-regular.
Denote by G the graph with V ( G ) = V ( G 1 × P 2 ) ( j { w j } ) , where G 1 is a graph as above and w j are additional vertices, and E ( G ) = E ( G 1 × P 2 ) ( j { [ w , w j ] : π 1 ( w ) N j } ) .
Lemma 8.
Let G 1 be a connected graph as above and P 2 with V ( P 2 ) = { v 1 , v 2 } . If G 1 is hyperbolic and B J as above is M-regular, then there exists a quasi-isometry f : G 1 × P 2 G with f ( w j , v i ) = w j for every j J and i { 1 , 2 } .
Proof. 
Let F : V ( G 1 × P 2 ) V ( G ) defined as F ( v , v i ) = ( v , v i ) for every v V ( G 1 ) , and F ( v , v i ) = w j for every v V ( B j ) . It is clear that F : V ( G 1 × P 2 ) V ( G ) is 0-full. Recall that we denote by π 1 : G 1 × P 2 G 1 the projection map. Define π : G G 1 as π = π 1 on G 1 × P 2 and π ( x ) = w j for every x with d G ( x , w j ) < 1 for some j J .
Now, we focus on proving that F : V ( G 1 × P 2 ) V ( G ) is a quasi-isometric embedding. For every ( w , v i ) , ( w , v i ) V ( G 1 × P 2 ) , one can check
d G ( F ( w , v i ) , F ( w , v i ) ) d G 1 × P 2 ( ( w , v i ) , ( w , v i ) ) .
To prove the other inequality, let us fix ( w , v i ) , ( w , v i ) V ( G 1 × P 2 ) (the inequalities in other cases can be obtained from the one in this case, as in the proof of Lemma 7). Consider a geodesic γ : = [ F ( w , v i ) F ( w , v i ) ] in G . If L ( γ ) = d G 1 × P 2 ( ( w , v i ) , ( w , v i ) ) , then
d G ( F ( w , v i ) , F ( w , v i ) ) = d G 1 × P 2 ( ( w , v i ) , ( w , v i ) ) .
If L ( γ ) < d G 1 × P 2 ( ( w , v i ) , ( w , v i ) ) , then π ( γ ) meets some B j . Since γ is a compact set, π ( γ ) intersects just a finite number of B j ’s, which we denote by B j 1 , B j r . We consider γ as an oriented curve from F ( w , v i ) to F ( w , v i ) ; thus we can assume that π ( γ ) meets B j 1 , , B j r in this order.
Let us define the following set of vertices in γ
{ w i 1 , w i 2 } : = γ ( N j i × P 2 ) ,
for every 1 i r , such that d G 1 × P 2 ( ( w , v i ) , w i 1 ) < d G 1 × P 2 ( ( w , v i ) , w i 2 ) . Note that [ w i 2 w i + 1 1 ] G 1 × P 2 for every 1 i < r and d G 1 × P 2 ( w i 2 , w i + 1 1 ) 1 since B ¯ j i B ¯ j i + 1 = .
If d G 1 ( π ( w i 1 ) , π ( w i 2 ) ) = d G 1 × P 2 ( w i 1 , w i 2 ) for some 1 i r , then d G 1 × P 2 ( w i 1 , w i 2 ) 2 K . Since d G 1 × P 2 ( w i 2 , w i + 1 1 ) 1 for 1 i < r , we have that d G 1 × P 2 ( w i 1 , w i 2 ) 2 K d G 1 × P 2 ( w i 2 , w i + 1 1 ) in this case.
If d G 1 ( π 1 ( w i 1 ) , π 1 ( w i 2 ) ) < d G 1 × P 2 ( w i 1 , w i 2 ) for some 1 i r , then d G 1 ( π 1 ( w i 1 ) , π 1 ( w i 2 ) ) + d G 1 × P 2 ( w i 1 , w i 2 ) is odd.
Since B J is M-regular, consider an odd cycle C with C B j i and L ( C ) < M , and let b i C B j i and [ π 1 ( w i 1 ) b i ] , [ b i π 1 ( w i 2 ) ] geodesics in G 1 . Thus, [ π 1 ( w i 1 ) b i ] [ b i π 1 ( w i 2 ) ] and [ π 1 ( w i 1 ) b i ] C [ b i π 1 ( w i 2 ) ] have different parity which means that one of them has different parity from [ π 1 ( w i 1 ) π 1 ( w i 2 ) ] . Then, d G 1 × P 2 ( w i 1 , w i 2 ) L ( [ π 1 ( w i 1 ) b i ] C [ b i π 1 ( w i 2 ) ] ) 4 K + M . Since d G 1 × P 2 ( w i 2 , w i + 1 1 ) 1 for 1 i < r , we have that d G 1 × P 2 ( w i 1 , w i 2 ) 4 K + M d G 1 × P 2 ( w i 2 , w i + 1 1 ) in this case.
Thus, we have that d G 1 × P 2 ( w i 1 , w i 2 ) 4 K + M for every 1 i r and d G 1 × P 2 ( w i 1 , w i 2 ) 4 K + M d G 1 × P 2 ( w i 2 , w i + 1 1 ) for every 1 i < r . Therefore,
d G 1 × P 2 ( ( w , v i ) , ( w , v i ) ) d G 1 × P 2 ( ( w , v i ) , w 1 1 ) + i = 1 r d G 1 × P 2 ( w i 1 , w i 2 ) + i = 1 r 1 d G 1 × P 2 ( w i 2 , w i + 1 1 ) + d G 1 × P 2 ( w r 2 , ( w , v i ) ) d G 1 × P 2 ( ( w , v i ) , w 1 1 ) + d G 1 × P 2 ( w r 2 , ( w , v i ) ) + 4 K + M + 1 i = 1 r 1 d G 1 × P 2 ( w i 2 , w i + 1 1 ) + d G 1 × P 2 ( w r 1 , w r 2 ) = d G ( F ( w , v i ) , F ( w 1 1 ) ) + d G ( F ( w r 2 ) , F ( w , v i ) ) + 4 K + M + 1 i = 1 r 1 d G ( F ( w i 2 ) , F ( w i + 1 1 ) ) + d G 1 × P 2 ( w r 1 , w r 2 ) 4 K + M + 1 ( d G ( F ( w , v i ) , F ( w 1 1 ) ) + d G ( F ( w r 2 ) , F ( w , v i ) ) + i = 1 r 1 d G ( F ( w i 2 ) , F ( w i + 1 1 ) ) ) + 4 K + M 4 K + M + 1 d G ( F ( w , v i ) , F ( w , v i ) ) + 4 K + M .
We conclude that F : V ( G 1 × P 2 ) V ( G ) is a quasi-isometric embedding. Thus, Lemma 1 provides a quasi-isometry f : G 1 × P 2 G with the required property. □
Definition 4.
Given a geodesic metric space X and closed connected pairwise disjoint subsets { η j } j J of X, we consider another copy X of X. The double D X of X is the union of X and X obtained by identifying the corresponding points in each η j and η j .
Definition 5.
Let us consider H > 0 , a metric space X, and subsets Y , Z X . The set V H ( Y ) : = { x X : d ( x , Y ) H } is called the H-neighborhood of Y in X. The Hausdorff distance of Y to Z is defined by H ( Y , Z ) : = inf { H > 0 : Y V H ( Z ) , Z V H ( Y ) } .
The following results in [15,80] will be useful.
Theorem 13.
([80], Theorem 3.2) Let us consider a geodesic metric space X and closed connected pairwise disjoint subsets { η j } j J of X, such that the double D X is a geodesic metric space. Then, the following conditions are equivalent:
(1)
D X is hyperbolic.
(2)
X is hyperbolic and there exists a constant c 1 such that for every k , l J and a η k , b η l we have d X ( x , j J η j ) c 1 for every x [ a b ] X .
(3)
X is hyperbolic and there exist constants c 2 , α , β such that for every k , l J and a η k , b η l we have d X ( x , j J η j ) c 2 for every x in some ( α , β ) -quasi-geodesic joining a with b in X.
Theorem 14.
([15], p. 87) For each δ 0 , a 1 and b 0 , there exists a constant H = H ( δ , a , b ) with the following property:
Let us consider a δ-hyperbolic geodesic metric space X and an ( a , b ) -quasigeodesic g starting in x and finishing in y. If γ is a geodesic joining x and y, then H ( g , γ ) H .
This property is called geodesic stability. It is well-known that hyperbolicity is, in fact, equivalent to geodesic stability [81].
Theorem 15.
Let G 1 be a connected graph and B j : = B G 1 ( w j , K j ) with w j V ( G 1 ) and K j Z + , for any j J , such that sup j K j = K < , B ¯ j 1 B ¯ j 2 = if j 1 j 2 , and every odd cycle C in G 1 satisfies C B j for some j J . Suppose { B j } j J is M-regular for some M > 0 . Let G 2 be a non-trivial bounded connected graph without odd cycles. Then, the following statements are equivalent:
(1)
G 1 × G 2 is hyperbolic.
(2)
G 1 is hyperbolic and there exists a constant c 1 , such that for every k , l J and w k B k , w l B l there exists a geodesic [ w k w l ] in G 1 with d G 1 ( x , j J w j ) c 1 for every x [ w k w l ] .
(3)
G 1 is hyperbolic and there exist constants c 2 , α , β , such that for every k , l J we have d G 1 ( x , j J w j ) c 2 for every x in some ( α , β )-quasi-geodesic joining w k with w l in G 1 .
Proof. 
Items ( 2 ) and ( 3 ) are equivalent by geodesic stability in G 1 (see Theorem 14).
Assume that ( 2 ) holds. By Lemma 7, there exists an ( α , β ) -quasi-isometry f : G 1 G 1 with f ( w j ) = w j for every j J . Given k , l J , f ( [ w k w l ] ) is an ( α , β ) -quasi-geodesic with endpoints w k and w l in G 1 . Given x f ( [ w k w l ] ) , we have x = f ( x 0 ) with x 0 [ w k w l ] and d G 1 ( x , j J w j ) α d G 1 ( x 0 , j J w j ) + β α c 1 + β . Taking X = G 1 , D X = G and η j = w j for every j J , Theorem 13 gives that G is hyperbolic. Now, Lemma 8 gives that G 1 × P 2 is hyperbolic and we conclude that G 1 × G 2 is hyperbolic by Lemma 2.
Now, suppose ( 1 ) holds. By Lemma 2, G 1 × P 2 is hyperbolic and, by Theorem 9, G 1 is hyperbolic. Then, Lemma 8 gives that G is hyperbolic and taking X = G 1 , D X = G and η j = w j for every j J , by Theorem 13, ( 2 ) holds. □
Theorem 15 and Remark 2 have the following consequence.
Corollary 5.
Let G 1 be a connected graph and suppose that there are a positive integer K and a vertex w G 1 , such that every odd cycle in G 1 intersects the open ball B : = B G 1 ( w , K ) . Let G 2 be a non-trivial bounded connected graph without odd cycles. Then, G 1 × G 2 is hyperbolic if and only if G 1 is hyperbolic.

4. Bounds for the Hyperbolicity Constant of Some Direct Products

The following well-known result will be useful (see a proof, e.g., in ([47], Theorem 8)).
Theorem 16.
In any connected graph G the inequality δ ( G ) ( diam G ) / 2 holds.
Remark 3.
Note that, if G 1 is a bipartite connected graph, then diam G 1 = diam V ( G 1 ) . Furthermore, if G 2 is a bipartite connected graph, then the product G 1 × G 2 has exactly two connected components, which are denoted by ( G 1 × G 2 ) 1 and ( G 1 × G 2 ) 2 , where each one is a bipartite graph and, consequently, diam ( G 1 × G 2 ) i = diam V ( ( G 1 × G 2 ) i ) for i { 1 , 2 } .
Remark 4.
Let P m , P n be two path graphs with m n 2 . The product P m × P n has exactly two connected components, which will be denoted by ( P m × P n ) 1 and ( P m × P n ) 2 . If u , v V ( ( P m × P n ) i ) for i { 1 , 2 } , then d ( P m × P n ) i ( u , v ) = max d P m ( π 1 ( u ) , π 1 ( v ) ) , d P n ( π 2 ( u ) , π 2 ( v ) ) and diam ( P m × P n ) i = diam V ( ( P m × P n ) i ) = m 1 .
Furthermore, if m 1 m and n 1 n , then δ ( P m × P n ) δ ( P m 1 × P n 1 ) .
Lemma 9.
Let P m , P n be two path graphs with m n 3 , and let γ be a geodesic in P m × P n such that there are two different vertices u , v in γ, with π 1 ( u ) = π 1 ( v ) . Then, L ( γ ) n 1 .
Remark 5.
Note that the conclusion of Lemma 9 does not hold for n = 2 , since we always have L ( γ ) 2 .
Proof. 
Let γ : = [ x y ] , and let V ( P m ) = { v 1 , , v m } , V ( P n ) = { w 1 , , w n } be the sets of vertices in P m , P n , respectively, such that [ v j , v j + 1 ] E ( P m ) and [ w i , w i + 1 ] E ( P n ) for 1 j < m , 1 i < n . Seeking for a contradiction, assume that L ( γ ) > n 1 . Notice that if [ u v ] denotes the geodesic contained in γ joining u and v, then π 2 restricted to [ u v ] is injective. Consider two vertices u , v γ such that [ u v ] [ u v ] γ , π 2 is injective in [ u v ] and π 2 ( u ) = w i 1 , π 2 ( v ) = w i 2 with i 2 i 1 maximal under these conditions. See Figure 4.
Since L ( γ ) > n 1 i 2 i 1 , either there is an edge [ v , w ] in G 1 × G 2 such that [ v , w ] ( γ \ [ u v ] ) or there is an edge [ u , w ] in G 1 × G 2 such that [ u , w ] ( γ \ [ u v ] ) . In addition, since L ( γ ) > n 1 , notice that π 2 is not injective in γ . Moreover, since i 2 i 1 is maximal, if π 2 ( w ) = w i 2 + 1 , then w γ , and since L ( γ ) > n 1 , u { x , y } and π 2 ( w ) = w i 1 + 1 . Thus, either π 2 ( w ) = w i 2 1 or π 2 ( w ) = w i 1 + 1 .
Hence, let us assume that there is an edge [ v , w ] in G 1 × G 2 such that [ v , w ] ( γ \ [ u v ] ) with π 2 ( w ) = w i 2 1 (otherwise, if there is an edge [ u , w ] in G 1 × G 2 such that [ u , w ] ( γ \ [ u v ] ) with π 2 ( w ) = w i 1 + 1 , the proof is similar).
Suppose π 1 ( v ) = v j . Let v be the vertex in [ u v ] such that π 2 ( v ) = w i 2 1 . Then, by construction of G 1 × G 2 , since v w , it follows that { π 1 ( v ) , π 1 ( w ) } = { v j 1 , v j + 1 } . Therefore, in particular, 1 < j < m .
Assume that v = ( v j 1 , w i 2 1 ) (if v = ( v j + 1 , w i 2 1 ) , then the argument is similar). Therefore, w = ( v j + 1 , w i 2 1 ) .
Consider the geodesic
σ = [ ( v j + 1 , w i 2 1 ) , ( v j , w i 2 2 ) ] [ ( v j , w i 2 2 ) , ( v j 1 , w i 2 3 ) ] [ ( v j 1 , w i 2 3 ) , ( v j 2 , w i 2 4 ) ]
Since π 1 ( u ) = π 1 ( v ) , there is a vertex ξ of V ( P m × P n ) in [ u v ] σ . Let s [ v , w ] γ with s v . Let σ 0 be the geodesic contained in σ joining ξ and w. Let γ 0 be the geodesic contained in γ joining ξ and s. Hence, L ( σ 0 [ w s ] ) < L ( σ 0 ) + 1 < L ( γ 0 ) leading to contradiction. □
Theorem 17.
Let P m , P n be two path graphs with m n 2 . If n = 2 , then δ ( P m × P 2 ) = 0 . If n 3 , then
min m 2 , n 1 1 δ ( P m × P n ) min m 2 , n 1 2 .
Furthermore, if m 2 n 3 and m is odd, then δ ( P m × P n ) = ( m 1 ) / 2 .
Proof. 
If m 2 , then P m × P 2 has two connected components isomorphic to P m , and δ ( P m × P 2 ) = 0 .
Assume that n 3 . By symmetry, it suffices to prove the inequalities for δ ( ( P m × P n ) 1 ) . Hence, Theorem 16 and Remark 4 give δ ( ( P m × P n ) 1 ) m 1 2 . By Theorem 7, there exists a geodesic triangle T = { x , y , z } T 1 ( P m × P n ) with p γ 1 : = [ x y ] , γ 2 : = [ x z ] , γ 3 : = [ y z ] , and δ ( ( P m × P n ) 1 ) = δ ( T ) = d ( P m × P n ) 1 ( p , γ 2 γ 3 ) . Let u V ( γ 1 ) such that d ( P m × P n ) 1 ( p , u ) 1 / 2 .
To prove δ ( ( P m × P n ) 1 ) n 1 / 2 , we consider two cases.
Assume first that there is at least a vertex v V ( ( P m × P n ) 1 ) T \ { u } such that π 1 ( u ) = π 1 ( v ) . If v γ 1 , then v γ 2 γ 3 and
δ ( T ) = d ( P m × P n ) 1 ( p , γ 2 γ 3 ) 1 / 2 + d ( P m × P n ) 1 ( u , v ) n 1 / 2 .
If v γ 1 , then L ( γ 1 ) n 1 by Lemma 9, and
δ ( T ) = d ( P m × P n ) 1 ( p , γ 2 γ 3 ) d ( P m × P n ) 1 ( p , { x , y } ) ( n 1 ) / 2 < n 1 / 2 .
Assume now that there is not a vertex v V ( ( P m × P n ) 1 ) T \ { u } such that π 1 ( u ) = π 1 ( v ) . Then, there exist two different vertices v 1 , v 2 in T \ { u } such that d ( P m × P n ) 1 ( u , v 1 ) = d ( P m × P n ) 1 ( u , v 2 ) = 1 , and π 1 ( v 1 ) = π 1 ( v 2 ) . If v 1 or v 2 belongs to γ 2 γ 3 , then δ ( T ) = d ( P m × P n ) 1 ( p , γ 2 γ 3 ) 3 / 2 n 1 / 2 . Otherwise, v 1 , v 2 γ 1 \ { u } . Lemma 9 gives L ( γ 1 ) n 1 , and we have that
δ ( T ) = d ( P m × P n ) 1 ( p , γ 2 γ 3 ) d ( P m × P n ) 1 ( p , { x , y } ) ( n 1 ) / 2 < n 1 / 2 .
To prove the lower bound, denote the vertices of P m and P n by V ( P m ) = { w 1 , w 2 , w 3 , , w m } and V ( P n ) = { v 1 , v 2 , v 3 , , v n } , with [ w i , w i + 1 ] E ( P m ) for 1 i < m and [ v i , v i + 1 ] E ( P n ) for 1 i < n .
Let ( P m × P n ) 1 be the connected component of P m × P n containing ( w 1 , v n 1 ) .
Assume first that m 2 n 3 . Consider the following curves in ( P m × P n ) 1 :
γ 1 : = [ ( w 1 , v n 1 ) , ( w 2 , v n ) ] [ ( w 2 , v n ) , ( w 3 , v n 1 ) ] [ ( w 3 , v n 1 ) , ( w 4 , v n ) ] [ ( w 2 n 4 , v n ) , ( w 2 n 3 , v n 1 ) ] , γ 2 : = [ ( w 1 , v n 1 ) , ( w 2 , v n 2 ) ] [ ( w 2 , v n 2 ) , ( w 3 , v n 3 ) ] [ ( w n 2 , v 2 ) , ( w n 1 , v 1 ) ] [ ( w n 1 , v 1 ) , ( w n , v 2 ) ] [ ( w 2 n 4 , v n 2 ) , ( w 2 n 3 , v n 1 ) ] .
Corollary 1 gives that γ 1 , γ 2 are geodesics. If B is the geodesic bigon B = { γ 1 , γ 2 } , then Remark 4 gives that
δ ( P m × P n ) δ ( B ) d ( P m × P n ) 1 ( ( w n 1 , v 1 ) , γ 1 ) = n 2 .
If m is odd with m 2 n 3 , then n ( m + 1 ) / 2 1 and we can consider the curves in ( P m × P n ) 1 :
γ 1 : = [ ( w 1 , v n 1 ) , ( w 2 , v n ) ] [ ( w 2 , v n ) , ( w 3 , v n 1 ) ] [ ( w 3 , v n 1 ) , ( w 4 , v n ) ] [ ( w m 1 , v n ) , ( w m , v n 1 ) ] , γ 2 : = [ ( w 1 , v n 1 ) , ( w 2 , v n 2 ) ] [ ( w 2 , v n 2 ) , ( w 3 , v n 3 ) ] [ ( w ( m + 1 ) / 2 1 , v n ( m + 1 ) / 2 + 1 ) , ( w ( m + 1 ) / 2 , v n ( m + 1 ) / 2 ) ] [ ( w ( m + 1 ) / 2 , v n ( m + 1 ) / 2 ) , ( w ( m + 1 ) / 2 + 1 , v n ( m + 1 ) / 2 + 1 ) ] [ ( w m 1 , v n 2 ) , ( w m , v n 1 ) ] .
Corollary 1 gives that γ 1 , γ 2 are geodesics. If B = { γ 1 , γ 2 } , then Remark 4 gives that
δ ( P m × P n ) δ ( B ) d ( P m × P n ) 1 ( ( w ( m + 1 ) / 2 , v n ( m + 1 ) / 2 ) , γ 1 ) = ( m 1 ) / 2 .
By Remark 4, if m is even with m 1 2 n 3 , then we have that
δ ( P m × P n ) δ ( P m 1 × P n ) ( m 2 ) / 2 .
Hence,
δ ( P m × P n ) n 2 , i f m 2 n 3 ( m 2 ) / 2 , i f m 2 n 2 = min n 2 , m 2 2 = min m 2 , n 1 1 .
Furthermore, if m 2 n 3 and m is odd, then we have proven ( m 1 ) / 2 δ ( P m × P n ) ( m 1 ) / 2 . □
Theorem 18.
If G 1 and G 2 are bipartite connected graphs with k 1 : = diam V ( G 1 ) and k 2 : = diam V ( G 2 ) such that k 1 k 2 1 , then
max min k 1 1 2 , k 2 1 , δ ( G 1 ) , δ ( G 2 ) δ ( G 1 × G 2 ) k 1 2 .
Furthermore, if k 1 2 k 2 2 and k 1 is even, then δ ( G 1 × G 2 ) = k 1 / 2 .
Proof. 
Corollary 1, Theorem 16 and Remark 3 give us the upper bound.
To prove the lower bound, we can see that there exist two path graphs P k 1 + 1 , P k 2 + 1 which are isometric subgraphs of G 1 and G 2 , respectively. It is easy to check that P k 1 + 1 × P k 2 + 1 is an isometric subgraph of G 1 × G 2 . By Lemma 3 and Theorem 17, we have
min k 1 1 2 , k 2 1 δ ( P k 1 + 1 × P k 2 + 1 ) δ ( G 1 × G 2 ) .
Using a similar argument as above, we have δ ( P 2 × G 2 ) δ ( G 1 × G 2 ) and δ ( G 1 × P 2 ) δ ( G 1 × G 2 ) . Thus, since ( G 1 × P 2 ) i G 1 and ( P 2 × G 2 ) i G 2 for i { 1 , 2 } , we obtain the first statement.
Furthermore, if k 1 + 1 2 ( k 2 + 1 ) 3 and k 1 + 1 is odd, then Theorem 17 gives δ ( P k 1 + 1 × P k 2 + 1 ) = k 1 / 2 , and we conclude δ ( G 1 × G 2 ) = k 1 / 2 . □
The following result deals just with odd cycles since otherwise we can apply Theorem 18.
Theorem 19.
For every odd number m 3 and every n 2 ,
δ ( C m × P n ) = m / 2 , i f n 1 m , ( n 1 ) / 2 , i f m < n 1 < 2 m , m 1 / 2 , i f n 1 2 m .
Proof. 
Let V ( C m ) = { w 1 , , w m } and V ( P n ) = { v 1 , , v n } be the sets of vertices in C m and P n , respectively, such that [ w 1 , w m ] , [ w j , w j + 1 ] E ( C m ) and [ v i , v i + 1 ] E ( P n ) for j { 1 , , m 1 } , i { 1 , , n 1 } . Note that for 1 j , r m and 1 i , s n , we have d C m × P n ( w j , v i ) , ( w r , v s ) = max { | i s | , | j r | } , if | i s | | j r | ( mod 2 ) , or d C m × P n ( w j , v i ) , ( w r , v s ) = max { | i s | , m | j r | } , if | i s | ¬ | j r | ( mod 2 ) . Besides, we have diam ( C m × P n ) = diam V ( C m × P n ) , i.e., diam ( C m × P n ) = m if n 1 m , and diam ( C m × P n ) = n 1 if n 1 > m . Thus, by Theorem 16, we have
δ ( C m × P n ) m / 2 , i f n 1 m , ( n 1 ) / 2 , i f n 1 > m .
Assume first that n 1 m . Note that C m × P 2 C 2 m and C m × P n is an isometric subgraph of C m × P n , if n n . By Lemma 3, we have δ ( C m × P n ) δ ( C 2 m ) = m / 2 , and we obtain the result in this case.
Assume now that n 1 > m . Consider the geodesic triangle T in C m × P n defined by the following geodesics
γ 1 : = [ ( w 1 , v n ) , ( w 2 , v n 1 ) ] [ ( w 2 , v n 1 ) , ( w 3 , v n ) ] [ ( w 3 , v n ) , ( w 4 , v n 1 ) ] [ ( w m 1 , v n 1 ) , ( w m , v n ) ] , γ 2 : = [ ( w ( m + 1 ) / 2 , v 1 ) , ( w ( m 1 ) / 2 , v 2 ) ] [ ( w ( m 1 ) / 2 , v 2 ) , ( w ( m 3 ) / 2 , v 3 ) ] [ ( w 2 , v ( m 1 ) / 2 ) , ( w 1 , v ( m + 1 ) / 2 ) ] [ ( w 1 , v ( m + 1 ) / 2 ) , ( w m , v ( m + 3 ) / 2 ) ] [ ( w m , v ( m + 3 ) / 2 ) , ( w 1 , v ( m + 5 ) / 2 ) ] [ ( w 1 , v ( m + 5 ) / 2 ) , ( w m , v ( m + 7 ) / 2 ) ] , γ 3 : = [ ( w ( m + 1 ) / 2 , v 1 ) , ( w ( m + 3 ) / 2 , v 2 ) ] [ ( w ( m + 3 ) / 2 , v 2 ) , ( w ( m + 5 ) / 2 , v 3 ) ] [ ( w m 1 , v ( m 1 ) / 2 ) , ( w m , v ( m + 1 ) / 2 ) ] [ ( w m , v ( m + 1 ) / 2 ) , ( w 1 , v ( m + 3 ) / 2 ) ] [ ( w 1 , v ( m + 3 ) / 2 ) , ( w m , v ( m + 5 ) / 2 ) ] [ ( w m , v ( m + 5 ) / 2 ) , ( w 1 , v ( m + 7 ) / 2 ) ] ,
where ( w 1 , v n ) (respectively, ( w m , v n ) ) is an endpoint of either γ 2 or γ 3 , depending of the parity of n. Since T is a geodesic triangle in C m × P n , we have δ ( C m × P n ) δ ( T ) . If n 1 < 2 m and M is the midpoint of the geodesic γ 3 , then δ ( C m × P n ) δ ( T ) = d C m × P n ( M , γ 1 γ 2 ) = L ( γ 3 ) / 2 = ( n 1 ) / 2 . Therefore, the result for m < n 1 < 2 m follows.
Finally, assume that n 1 2 m . Let us consider N γ 3 such that d C m × P n N , ( w ( m + 1 ) / 2 , v 1 ) = m 1 / 2 . Thus, δ ( C m × P n ) δ ( T ) d C m × P n ( N , γ 1 γ 2 ) = d C m × P n N , ( w ( m + 1 ) / 2 , v 1 ) = m 1 / 2 . To finish the proof, it suffices to prove that δ ( C m × P n ) m 1 / 2 . Seeking for a contradiction, assume that δ ( C m × P n ) > m 1 / 2 . By Theorems 6 and 7, there is a geodesic triangle = { x , y , z } T 1 ( C m × P n ) and p [ x y ] with d C m × P n ( p , [ y z ] [ z x ] ) = δ ( C m × P n ) m 1 / 4 . Then, L ( [ x y ] ) = d C m × P n ( x , p ) + d C m × P n ( p , y ) 2 m 1 / 2 . Let V x (respectively, V y ) be the closest vertex to x (respectively, y) in [ x y ] , and consider a vertex V p in [ x y ] such that d C m × P n p , V ( C m × P n ) = d C m × P n ( p , V p ) . Note that d C m × P n ( p , [ y z ] [ z x ] ) m 1 / 4 implies that d C m × P n ( p , V p ) 1 / 2 . Since x , y , z J ( C m × P n ) , we have d C m × P n ( V x , V y ) 2 m 1 > m and, consequently, π 2 ( [ x y ] ) is a geodesic in P n . Since π 2 ( [ y z ] [ z x ] ) is a path in P n joining π 2 ( x ) and π 2 ( y ) , there exists a vertex ( u , v ) [ x z ] [ z y ] such that π 2 ( V p ) = v and u π 1 ( V p ) . Therefore, d C m × P n V p , ( u , v ) m 1 and, consequently, d C m × P n ( p , [ x z ] [ z y ] ) d C m × P n ( p , V p ) + d C m × P n ( V p , [ x z ] [ z y ] ) 1 / 2 + m 1 , leading to contradiction. □

5. Conclusions

In this paper, we characterize in many cases the hyperbolic direct product of graphs. Here, the situation is more complex than with other graph products, partly because the direct product of two bipartite graphs is already disconnected and the formula for the distance in G 1 × G 2 is more complicated than in the case of other products of graphs. Although in the study of hyperbolicity in a general context the hypothesis on the existence (or non-existence) of odd cycles is artificial, in the study of hyperbolic direct products, it is an essential hypothesis. We have proven that, if G 1 × G 2 is hyperbolic, then one factor is hyperbolic and the other one is bounded. Besides, we prove that this necessary condition is also sufficient in many cases. If G 1 is a hyperbolic graph and G 2 is a bounded graph, then we prove that G 1 × G 2 is hyperbolic when G 2 has some odd cycle or G 1 and G 2 do not have odd cycles. Otherwise, the characterization of hyperbolic direct products is a more difficult task. If G 1 has some odd cycle and G 2 does not have odd cycles, we provide sufficient conditions for non-hyperbolicity and hyperbolicity, respectively. Besides, we characterize the hyperbolicity of G 1 × G 2 under some additional conditions.
A natural open problem is the complete characterization of hyperbolic direct products.
A second open problem is to compute the precise value of the hyperbolicity constant of the graphs appearing in Theorems 17 and 18 with unknown hyperbolicity constant.
Direct product of graphs is a subject closely related to lift of graphs, which have been intensively studied (see, e.g., [82] and the references therein). Another interesting problem is to study the hyperbolicity of lift of graphs. We think that it is possible to obtain some similar results in this context, although the odd cycles may not play an important role in the study of hyperbolic lifts of graphs.

Author Contributions

The authors contributed equally to this work.

Funding

This work was supported in part by four grants from Ministerio de Economía y Competititvidad (MTM2012-30719, MTM2013-46374-P, MTM2016-78227-C2-1-P and MTM2015-69323-REDT), Spain.

Acknowledgments

We thank the referees for their suggestions and helpful remarks.

Conflicts of Interest

The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.

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Figure 1. If G 1 and G 2 are unbounded, for any odd n, there is a geodesic triangle T G 1 × G 2 with δ ( T ) n 3 2 .
Figure 1. If G 1 and G 2 are unbounded, for any odd n, there is a geodesic triangle T G 1 × G 2 with δ ( T ) n 3 2 .
Symmetry 10 00279 g001
Figure 2. Two geodesic triangles, T 1 , T 2 , which are odd cycles and a geodesic g joining them define an even closed path.
Figure 2. Two geodesic triangles, T 1 , T 2 , which are odd cycles and a geodesic g joining them define an even closed path.
Symmetry 10 00279 g002
Figure 3. If d G 1 ( m , T 1 g ) 8 δ , then m [ x z ] and there is a point m [ x m ] [ x w k ] such that d G 1 ( m , m ) = 2 ( D + 8 δ ) .
Figure 3. If d G 1 ( m , T 1 g ) 8 δ , then m [ x z ] and there is a point m [ x m ] [ x w k ] such that d G 1 ( m , m ) = 2 ( D + 8 δ ) .
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Figure 4. For any geodesic γ in P m × P n with π 1 ( u ) = π 1 ( v ) for some different vertices u , v in γ , then L ( γ ) n 1 .
Figure 4. For any geodesic γ in P m × P n with π 1 ( u ) = π 1 ( v ) for some different vertices u , v in γ , then L ( γ ) n 1 .
Symmetry 10 00279 g004

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