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 hc(G) to be the least integerm such that the iterated line graph Lm(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 ≤ hc(G) ≤ max{diam(G), k − 1}. We also show that κ3(G) ≤ hc(G) ≤ κ3(G) + 2 where κ3(G) is the least integer m such that Lm(G) is 3-connected. Finally we prove that hc(G) ≤ |V (G)| −∆(G)+ 1. These bounds are all sharp. © 2008 Elsevier B.V. All rights reserved.