In 1956, Tutte proved that a 4-connected planar graph is Hamiltonian. Moreover, in 1997, Sanders extended this to the result that a 4-connected planar graph contains a Hamiltonian cycle through any two of its edges. Harant and Senitsch [J. Harant, S. Senitsch, A generalization of Tutte’s theorem on Hamiltonian cycles in planar graphs, Discrete Mathematics 309 (2009) 4949–4951] even proved that a planar graph has a cycle containing a given subset of its vertex set and any two prescribed edges of the subgraph of induced by if and if is 4-connected in . If , then Sanders’ result follows.
Here, we consider the case that is 5-connected in and that there are prescribed edges and forbidden edges of for a cycle through .
Highlights
► A 5-connected set of vertices of a planar graph is considered. ► It is known that contains a cycle through . ► The options to prescribe edges and to forbid edges for are discussed.