Prescribed edges and forbidden edges for a cycle in a planar graph

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 GG has a cycle containing a given subset XX of its vertex set and any two prescribed edges of the subgraph G[X]G[X] of GG induced by XX if |X|≥3|X|≥3 and if XX is 4-connected in GG. If X=V(G)X=V(G), then Sanders’ result follows.Here, we consider the case that XX is 5-connected in GG and that there are prescribed edges and forbidden edges of G[X]G[X] for a cycle through XX.

► A 5-connected set XX of vertices of a planar graph GG is considered. ► It is known that GG contains a cycle CC through XX. ► The options to prescribe edges and to forbid edges for CC are discussed.