Radius and subpancyclicity in line graphs

A graph is called subpancyclic if it contains cycles of length from 3 to its circumference. Let G be a graph with min {d (u) + d (v) : u v ∈ E (G)} ≥ 8. In this paper, we prove that if one of the following holds: the radius of G is at most ⌊ frac(Δ (G), 2) ⌋; G has no subgraph isomorphic to Y_{Δ (G) + 2}; the circumference of G is at most Δ (G) + 1; the length of a longest path is at most Δ (G) + 1, then the line graph L (G) is subpancyclic and these conditions are all best possible even under the condition that L (G) is hamiltonian.