Hamilton-connected indices of graphs

Let GG 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)hc(G) to be the least integer mm such that the iterated line graph Lm(G)Lm(G) is Hamilton-connected. Let diam(G) be the diameter of GG and kk be the length of a longest path whose internal vertices, if any, have degree 2 in GG. 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)+2κ3(G)≤hc(G)≤κ3(G)+2 where κ3(G)κ3(G) is the least integer mm such that Lm(G)Lm(G) is 3-connected. Finally we prove that hc(G)≤|V(G)|−Δ(G)+1hc(G)≤|V(G)|−Δ(G)+1. These bounds are all sharp.