The nonorientable genus of $K_n$

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...

The Nonorientable Four-genus of Knots

We develop obstructions to a knot K ⊂ S bounding a smooth punctured Klein bottle in B. The simplest of these is based on the linking form of the 2–fold branched cover of S branched over K. Stronger obstructions are based on the Ozsváth-Szabó correction term in Heegaard-Floer homology, along with the G–signature theorem and the Guillou-Marin generalization of Rokhlin’s theorem. We also apply Cas...

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.

Nonorientable Genus of Nearly Complete Bipartite Graphs

Tutte's 5-flow conjecture for graphs of nonorientable genus 5

We develop four constructions for nowhere-zero 5-ows of 3-regular graphs which satisfy special structural conditions. Using these constructions we show a minimal counterexample to Tutte's 5-ow conjecture is of order 44 and therefore every bridgeless graph of nonorientable genus 5 has a nowhere-zero 5-ow. One of the structural properties is formulated in terms of the structure of the multigraph ...