In　1984,　Bauer　proposed　the　problems　of　determining　best　possible　sufficient　conditions　on　the　vertex　degrees　of　a　simple　graph　(or　a　simple　bipartite　graph,　or　a　simple　triangle-free　graph,　respectively)　G　to　ensure　that　its　line　graph　L(G)　is　hamiltonian.　We　investigate　the　problems　of　determining　best　possible　sufficient　conditions　on　the　vertex　degrees　of　a　simple　graph　G　to　ensure　that　its　line　graph　L(G)　is　hamiltonian-connected,　and　prove　the　following.　(i)　For　any　real　numbers　a,b　with　0<a<1,　there　exists　a　finite　family　F(a,b)　such　that　for　any　connected　simple　graph　G　on　n　vertices,　if　dG(u)+dG(v)≥an+b　for　any　u,v∈V(G)　with　uv∉E(G),　then　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　a　member　in　F(a,b).　(ii)　Let　G　be　a　connected　simple　graph　on　n　vertices.　If　dG(u)+dG(v)≥[Formula　presented]−2　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.　(iii)　Let　G　be　a　connected　simple　triangle-free　(or　bipartite)　graph　on　n　vertices.　If　dG(u)+dG(v)≥[Formula　presented]　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.　©　2018　Elsevier　B.V.