Power graphs of (non)orientable genus two

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...

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.

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 ...

The nonorientable genus of joins of complete graphs with large edgeless graphs

We show that for n = 4 and n ≥ 6, Kn has a nonorientable embedding in which all the faces are hamilton cycles. Moreover, when n is odd there is such an embedding that is 2-facecolorable. Using these results we consider the join of an edgeless graph with a complete graph, Km +Kn = Km+n − Km, and show that for n ≥ 3 and m ≥ n − 1 its nonorientable genus is d(m − 2)(n − 2)/2e except when (m,n) = (...