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.
Net Spaces in Categorical Topology a
In the paper we introduce the notion of S‐nets, S a nonempty construct, as a generalization of the usual nets. S‐net spaces and continuous maps between them are then defined in a natural way. For S‐net spaces we introduce a number of axioms and study the categories obtained. We find some topological and Cartesian closed categories of S‐net spaces and describe their relations to certain known topological categories.