Hyperbolicity of direct products of graphs

TL;DR

This paper investigates the hyperbolicity of direct product graphs, establishing conditions under which the product is hyperbolic based on properties of the factor graphs, and provides bounds for their hyperbolicity constants.

Contribution

It characterizes when the direct product of graphs is hyperbolic, showing one factor must be hyperbolic and the other bounded, and offers formulas or bounds for hyperbolicity constants.

Findings

If the direct product is hyperbolic, one factor is hyperbolic and the other bounded.

Necessary conditions for hyperbolicity are also sufficient in many cases.

Provides bounds and formulas for hyperbolicity constants of certain graph products.

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 $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ 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., $\delta(X)=\inf\{\delta\ge 0: \, X \, \text{ is $\delta$-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 $G_1\times G_2$ 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.