Gromov hyperbolicity in lexicographic product graphs

Abstract

If $X$ is a geodesic metric space and $x_1,x_2,x_3\in X$, a {\it 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 $\delta$-\emph{hyperbolic} $($in the Gromov sense$)$ if any side of $T$ is contained in a $\delta$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. If $X$ is hyperbolic, we denote by $\delta (X)$ the sharp hyperbolicity constant of $X$, i.e. $δ(X)=\inf\{δ\ge 0: \, X \, \text{ is $δ$-hyperbolic}\}.$ In this paper we characterize the lexicographic product of two graphs $G_1\circ G_2$ which are hyperbolic, in terms of $G_1$ and $G_2$: the lexicographic product graph $G_1\circ G_2$ 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 $G_1\circ G_2$ is hyperbolic if and only if $G_2$ is hyperbolic. In particular, we obtain the sharp inequalities $\delta (G_1)\le \delta (G_1\circ G_2)\le \delta (G_1)+3/2$ if $G_1$ is not a trivial graph, and we characterize the graphs for which the second inequality is attained.

arXiv admin note: text overlap with arXiv:1410.2938