After our summer break our regular study group resumed on Friday. This week, John started what looks like being a two week discussion about and generalised hexagons.
A generalised polygon (n-gon) of order (s,t) is a 1-(v,s+1,t+1) design (that is, a set of v points and a collection of (s+1)-subsets called blocks, or lines, such that each point lies in t+1 blocks) whose incidence graph has girth 2n and diameter n. A polygon with n vertices and the edges taken to be the blocks, is a generalised n-gon with s=t=1. The incidence graph of such a polygon is a cycle of length 2n. We call a generalised polygon thick if .
Some authors define a generalised n-gon to be a bipartite graph with girth 2n and diameter n. John does not like this definition as it is referring to the incidence graph and not the design/geometry. This reminds me of a question Alice, Gordon and I had a couple of weeks ago which we were not able to resolve. We were wondering if “bipartite” was necessary in this definition. That is, is a graph of girth 2n and diameter n bipartite? Any answers are more than welcome.
The incidence graph of a generalised 2-gon is a complete bipartite graph. The generalised 3-gons are the projective planes and the generalised 4-gons, or generalised quadrangles, are the rank two polar spaces.
Walter Feit and Graham Higman proved in 1964 (in the first volume of the Journal of Algebra) that finite thick generalised n-gons only exist for and . While there are many projective planes and generalised quadrangles, there are only two known infinite families of generalised hexagons and one known infinite family of generalised octagons.