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 G1×G2 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).
Highlights
An interesting topic in graph theory is the study of the different types of products of graphs [1].In particular, given two graphs G1, G2, the direct product G1 × G2 is defined as the graph with vertices the (Cartesian) product of V ( G1 ) and V ( G2 ), and two vertices (u1, v1 ), (u2, v2 ) ∈ V ( G1 × G2 ) are connected by an edge if and only if [u1, u2 ] ∈ E( G1 ) and [v1, v2 ] ∈ E( G2 )
If G1 is a hyperbolic graph and G2 is a bounded graph, we prove that G1 × G2 is hyperbolic when G2 has some odd cycle (Theorem 3) or G1 and
We characterize in many cases the hyperbolic direct product of graphs
Summary
An interesting topic in graph theory is the study of the different types of products of graphs [1]. 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 G1 × G2 is more complicated that in the case of other products of graphs The symmetry of this product allows us to show that, if G1 × G2 is hyperbolic, one factor is hyperbolic and the other one is bounded (see Theorem 10). 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 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 essential in the study of hyperbolic product graphs
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