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 other two sides, for every geodesic triangle T in X. An important problem in the study of hyperbolic graphs is to relate the hyperbolicity with some classical properties in graph theory. In this paper we find a very close connection between hyperbolicity and chordality: we extend the classical definition of chordality in two ways, edge-chordality and path-chordality, in order to relate this property with Gromov hyperbolicity. In fact, we prove that every edge-chordal graph is hyperbolic and that every hyperbolic graph is path-chordal. Furthermore, we prove that every path-chordal cubic graph with small path-chordality constant is hyperbolic.

If X is a geodesic metric space and x1, x2, x3 ∈ X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] 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. We denote by δ(X) the sharp hyperbolicity constant of X, i.e. δ(X) := inf{δ ≥ 0 : X is δ-hyperbolic } .

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 $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ 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 other two sides, for every geodesic triangle $T$ in $X$. We denote by $\delta(X)$ the sharp hyperbolicity constant of $X$, i.e., $\delta(X):=\inf\{\delta\ge 0: \, X \, \text{ is $\delta$-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. One of the main aims of this paper is to obtain quantitative information about the distortion of the hyperbolicity constant of the graph $G\setminus e$ obtained from the graph $G$ by deleting an arbitrary edge $e$ from it. These inequalities allow to obtain the other main result of this paper, which characterizes in a quantitative way the hyperbolicity of any graph in terms of local hyperbolicity.

Let G = (V, E) be a graph of order n and let 1 ≤ k < n be an integer. The k-token graph of G is the graph whose vertices are all the k-subsets of V, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. In this paper we characterize precisely, for each value of k, which graphs have a regular k-token graph and which connected graphs have a planar k-token graph.