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 κ 3 ( G ) ≤ h c ( G ) ≤ κ 3 ( G ) + 2 where κ 3 ( 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.