SYGN 2012 Abstracts - Steve Wilson

Tales From the Census
Steve Wilson (Northern Arizona University, USA)

One topic of interest in the production of the census of edge-transitive teravalent graphs are semitransitive orientations which are graded.  This means a directed graph whose group is transitive on vertices and (directed) edges for which there is a labelling of the vertices with numbers mod k (for some k at least 3) so that each edge points from a vertex labelled i to one labelled i+1.  All tightly attached orientations are graded but there are many graded orientations which are not tightly attached.

We will first describe a construction by Casey Attebery which gives lots of graded orientations.  We inserted many of these into the census, but, embarrassingly, a, um, programming flaw allowed into the census one example which fails the Attebery's construction, but is in the census anyway because it is, well, it is semisymmetric.  Moreover, no other construction we know gives this interesting graph.  How can we generalize it?   Where did it come from, really?  What is it like?