Hamilton-connected indices of graphs

Let G be an undirected graph that is neither a path nor a cycle. Clark and Wormald [L.H. Clark, N.C. Wormald, Hamiltonian-like indices of graphs, ARS Combinatoria 15 (1983) 131-148] defined h c (G) to be the least integer m such that the iterated line graph L^m (G) is Hamilton-connected. Let diam (G) be the diameter of G and k be the length of a longest path whose internal vertices, if any, have degree 2 in G. In this paper, we show that k - 1 ≤ h c (G) ≤ max {diam (G), k - 1}. We also show that κ³ (G) ≤ h c (G) ≤ κ³ (G) + 2 where κ³ (G) is the least integer m such that L^m (G) is 3-connected. Finally we prove that h c (G) ≤ | V (G) | - Δ (G) + 1. These bounds are all sharp.