Every3-connected Essentially9-connected Line Graphs Is Hamilton-connected

The Hamiltonian problem is an important research topic in structural graph theory. It is well known that the problem is closely related to Four Color Problem. The problem of deciding whether a given graph has a Hamiltonian cycle or not is a NP-hard by the issue of computational complexity. Therefore, the research of Hamiltonian problem is concentrated on sufficient implying a graph to be Hamil-tonian. This paper, we based on the subgraph structure, forbidding some specific subgraph. A graph is H-free if it contains no induced subgraph isomorphic to H, we also say H is forbidden in G. K1,3-free is called claw-free. As we all know, a line graph is a claw-free graph. Thomassen and Matthews conjectured that every4-connected line graph is hamiltonian in1984. This conjecture has been studied extensively for over twenty years (see [12],[4][15] and others), but it is still open. On the other hand, the Hamiltonicity of3-connected line graphs was firstly consid-ered by Lai et al., who proved that Every3-connected essentially11-connected line graphs is Hamiltonian in2006. Li et al. proved that Every3-connected essentially10-connected line graphs is Hamiltonian in2012. Li et al. asked what is the small-est integer k such that every3-edge-connected, essentially κ-edge-connected graph is Hamilton-connected. In this paper, we show that every3-connected, essentially9-connected line graphs is Hamilton-connected. This result extends the related work of Lai et al. and Li et al.