Hyperbolicity of Direct Products of Graphs

Abstract

If X is a geodesic metric space and x 1 , x 2 , x 3 ∈ X, a geodesic triangle T = {x 1 , x 2 , x 3 } is the union of the three geodesics [x 1 x 2 ], [x 2 x 3 ] and [x 3 x 1 ] 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. If X is hyperbolic, we denote by δ(X) the sharp hyperbolicity constant of X, i.e., δ(X) = inf{δ ≥ 0 : X is δ-hyperbolic }. Some previous works characterize the hyperbolic product graphs (for the Cartesian, strong, join, corona and lexicographic products) in terms of properties of the factor graphs. However, the problem with the direct product is more complicated. In this paper, we prove that if the direct product G 1 × G 2 is hyperbolic, then one factor is hyperbolic and the other one is bounded. Also, we prove that this necessary condition is, in fact, a characterization in many cases. In other cases, we find characterizations which are not so simple. Furthermore, we obtain formulae or good bounds for the hyperbolicity constant of the direct product of some important graphs.

Small values of the hyperbolicity constant in graphs

“…The invariance of the hyperbolicity under some natural transformations on graphs have been studied in previous papers, for instance, removing edges of a graph is studied in [14,28]. Moreover, the hyperbolicity of some product graphs have been characterized: in [23,24,25,26,27,30,67] the authors characterize in a simple way the hyperbolicity of strong product of graphs, direct product of graphs, lexicographic product of graphs, Cartesian sum of graphs, graph join and corona, and Cartesian product of graphs. Some other authors have obtained results on hyperbolicity for particular classes of graphs: chordal graphs, vertex-symmetric graphs, bipartite and intersection graphs, bridged graphs and expanders [20,94,66,22,42,61,65].…”