2-edge-Hamiltonian-connectedness of 4-connected plane graphs

A graph G is called 2-edge-Hamiltonian-connected if for any X ⊂ { x 1 x 2 : x 1 , x 2 ∈ V ( G ) } with 1 ≤ | X | ≤ 2 , G ∪ X has a Hamiltonian cycle containing all edges in X , where G ∪ X is the graph obtained from G by adding all edges in X . In this paper, we show that every 4-connected plane graph is 2-edge-Hamiltonian-connected. This result is best possible in many senses and an extension of several known results on Hamiltonicity of 4-connected plane graphs, for example, Tutte’s result saying that every 4-connected plane graph is Hamiltonian, and Thomassen’s result saying that every 4-connected plane graph is Hamiltonian-connected. We also show that although the problem of deciding whether a given graph is 2-edge-Hamiltonian-connected is N P -complete, there exists a polynomial time algorithm to solve the problem if we restrict the input to plane graphs.