Degree sum and hamiltonian-connected line graphs

Abstract

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)≥n4−2 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)≥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.