Degree sum and hamiltonian-connected line graphs

(iii) Let G be a connected simple triangle-free (or bipartite) graph on n vertices. If dG(u)+dG(v)≥n8 for any u,v∈V(G) with uv∉E(G), then for sufficiently large n, either L(G) is hamiltonian-connected, or κ(L(G))≤2, or L(G) is not hamiltonian-connected, κ(L(G))≥3 and G is contractible to W8, the Wagner graph.