Hyperbolicity and Chordality of a Graph

Let G be a connected graph with the usual shortest-path metric d. The graph G is δ-hyperbolic provided for any vertices x, y, u, v in it, the two larger of the three sums d(u, v) + d(x, y), d(u, x) + d(v, y) and d(u, y) + d(v, x) differ by at most 2δ. The graph G is k-chordal provided it has no induced cycle of length greater than k. Brinkmann, Koolen and Moulton find that every 3-chordal graph is 1-hyperbolic and that graph is not 12 -hyperbolic if and only if it contains one of two special graphs as an isometric subgraph. For every k ≥ 4, we show that a k-chordal graph must be ⌊ 2 ⌋ 2 -hyperbolic and there does exist a k-chordal graph which is not ⌊ 2 ⌋ 2 -hyperbolic. Moreover, we prove that a 5-chordal graph is 12 -hyperbolic if and only if it does not contain any of a list of five special graphs as an isometric subgraph.