Radius and subpancyclicity in line graphs

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