A generalization of Tutte's theorem on Hamiltonian cycles in planar graphs

In 1956, W.T. Tutte proved that a 4-connected planar graph is hamiltonian. Moreover, in 1997, D.P. Sanders extended this to the result that a 4-connected planar graph contains a hamiltonian cycle through any two of its edges. We prove that a planar graph G has a cycle containing a given subset X of its vertex set and any two prescribed edges of the subgraph of G induced by X if |X| ≥ 3 and if X is 4-connected in G. If X = V (G) then Sanders’ result follows. We also discuss the problem under which condition a 4-connected planar graph has a hamiltonian cycle through more than two specified edges. (Joint work with F. Göring and S. Senitsch.)