On the Hyperbolicity Constant of Line Graphs

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 }. 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 line graph L(G) in terms of parameters of the graph G. In particular, we prove qualitative results as the following: a graph G is hyperbolic if and only if L(G) is hyperbolic; if {Gn} is a T-decomposition of G ({Gn} are simple subgraphs of G), the line graph L(G) is hyperbolic if and only if supn δ(L(Gn)) is finite. Besides, we obtain quantitative results. Two of them are quantitative versions of our qualitative results. We also prove that g(G)/4 ≤ δ(L(G)) ≤ c(G)/4 + 2, the electronic journal of combinatorics 18 (2011), #P210 1 where g(G) is the girth of G and c(G) is its circumference. We show that δ(L(G)) ≥ sup{L(g) : g is an isometric cycle in G }/4. Furthermore, we characterize the graphs G with δ(L(G)) < 1.