Characterization of the hyperbolicity in the lexicographic product

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 } . In this paper we characterize the lexicographic product of two graphs G 1 ∘ G 2 which are hyperbolic, in terms of G 1 and G 2 : the lexicographic product graph G 1 ∘ G 2 is hyperbolic if and only if G 1 is hyperbolic, unless if G 1 is a trivial graph (the graph with a single vertex); if G 1 is trivial, then G 1 ∘ G 2 is hyperbolic if and only if G 2 is hyperbolic. In particular, we obtain that δ ( G 1 ) ≤ δ ( G 1 ∘ G 2 ) ≤ δ ( G 1 ) + 3 / 2 if G 1 is not a trivial graph, and we find families of graphs for which the inequalities are attained.