Hamilton cycles and closed trails in iterated line graphs

Let G be an undirected connected graph that is not a path. We define h(G) (respectively, s(G)) to be the least integer m such that the iterated line graph Lm(G) has a Hamiltonian cycle (respectively, a spanning closed trail). To obtain upper bounds on h(G) and s(G), we characterize the least integer m such that Lm(G) has a connected subgraph H, in which each edge of H is in a 3-cycle and V (H) contains all vertices of degree not 2 in Lm(G). We characterize the graphs G such that h(G) − 1 (respectively, s(G)) is greater than the radius of G.