A remark on embedded bipartite graphs
Abstract A use of Euler's formula in Zaks ( J. Combin. Theory Ser. B 32 (1982) , 95–98) is replaced by an elementary argument on permutations.
Extending an extension of Grinberg's theorem
Abstract Grinberg's well-known necessary condition for a planar graph to be Hamiltonian was extended by Shimamoto to a condition for having a 2-factor. We derive a necessary condition for a planar graph to contain a collection of n disjoint circuits, none of which separates any other two. All is done in terms of the numbers p k which denote the numbers of k -gonal faces.
On an extension of the Grinberg theorem
Abstract An extension of the Grinberg condition for the existence of Hamiltonian circuits in a planar graph is discussed.