One-Factorisations of Complete Graphs Constructed in Desarguesian Planes of Certain Odd Square Orders

A geometric construction of one-factorisations of complete graphs $K_{q(q-1)}$ is provided for the case when either $q=2^d +1$ is a Fermat prime, or $q=9$. This construction uses the affine group $\mathrm{AGL}(1,q)$, points and ovals in the Desarguesian plane $\mathrm{PG}(2,q^{2})$ to produce one-factorisations of the complete graph $K_{q(q-1)}$.

Non-Desarguesian planes and weak associativity

During the 19th century various criticisms of Euclid's geometry emerged and alternative axiom systems were constructed. That of David Hilbert ([1], 1899) paid particular attention to the independence of the axioms, and it is his insights which have shaped many of the further developments during the 20th century.We can, from his insights, define an affine plane as a set of points, with distinguished subsets called lines such thatAxiom 1: Given two distinct points, there is a unique line containing them both.Axiom 2: Given a line L and a point, p, not contained in L, there is a unique line containing p which does not intersect L.Axiom 3: There exist at least three points, not belonging to the same line.

On KM-Arcs in Small Desarguesian Planes

In this paper we study the existence problem for KM-arcs in small Desarguesian planes. We establish a full classification of KM$_{q,t}$-arcs for $q\le 32$, up to projective equivalence. We also construct a KM$_{64,4}$-arc; as $t=4$ was the only value for which the existence of a KM$_{64,t}$-arc was unknown, this fully settles the existence problem for $q\le 64$.

Arcs with Large Conical Subsets in Desarguesian Planes of Even Order

We give an explicit classification of the arcs in PG$(2,q)$ ($q$ even) with a large conical subset and excess 2, i.e., that consist of $q/2+1$ points of a conic and two points not on that conic. Apart from the initial setup, the methods used are similar to those for the case of odd $q$, published earlier (Electronic Journal of Combinatorics, 17, #R112).