Hyperbolicity and complement of graphs

Abstract

If X is a geodesic metric space and $x_1,x_2,x_3\in X$, a geodesic triangle $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $\left[x_1{x}_2\right]$, $\left[x_2{x}_3\right]$ and $\left[x_3{x}_1\right]$ in $X$. The space $X$ is $\delta$-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 $X$. We denote by $\delta \left(X\right)$ the sharp hyperbolicity constant of $X$, i.e. $\delta \left(X\right)≔inf\{\delta \ge 0:X\text{ is }\delta \text{-hyperbolic}\}$. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. The main aim of this paper is to obtain information about the hyperbolicity constant of the complement graph $\overline{G}$ in terms of properties of the graph $G$. In particular, we prove that if $\mathrm{diam}\left(V\left(G\right)\right)\ge 3$, then $\delta \left(\overline{G}\right)\le 2$, and that the inequality is sharp. Furthermore, we find some Nordhaus–Gaddum type results on the hyperbolicity constant of a graph $\delta \left(G\right)$.