Orientable and nonorientable genus of the complete bipartite graph

Nonorientable Genus of Nearly Complete Bipartite Graphs

On the orientable genus of graphs with bounded nonorientable genus

A conjecture of Robertson and Thomas on the orientable genus of graphs with a given nonorientable embedding is disproved.

The nonorientable genus of complete tripartite graphs

In 1976, Stahl and White conjectured that the nonorientable genus of Kl,m,n, where l ≥ m ≥ n, is (l−2)(m+n−2) 2 . The authors recently showed that the graphs K3,3,3 , K4,4,1, and K4,4,3 are counterexamples to this conjecture. Here we prove that apart from these three exceptions, the conjecture is true. In the course of the paper we introduce a construction called a transition graph, which is cl...

Orientable and nonorientable minimal surfaces

We describe a theory of Morse-Conley type for orientable and nonorientable minimal surfaces of varying topological type solving Plateau problems in R 3 . 1991 Mathematics Subject Classification 53A10, 49F10, 58E12, 32G 15 After J. Douglas had solved Plateau's problem (this was also independently achieved by T. Rad6), he considered the problem of finding minimal surfaces of higher genus and/or c...

Classification of nonorientable regular embeddings of complete bipartite graphs

A 2-cell embedding of a graph G into a closed (orientable or nonorientable) surface is called regular if its automorphism group acts regularly on the flags mutually incident vertex-edge-face triples. In this paper, we classify the regular embeddings of complete bipartite graphs Kn,n into nonorientable surfaces. Such a regular embedding of Kn,n exists only when n = 2p a1 1 p a2 2 · · · p ak k (a...