The diameter of random Cayley digraphs of given degree

We consider random Cayley digraphs of order $n$ with uniformly distributed generating set of size $k$. Specifically, we are interested in the asymptotics of the probability such a Cayley digraph has diameter two as $n\to\infty$ and $k=f(n)$. We find a sharp phase transition from 0 to 1 at around $k = \sqrt{n \log n}$. In particular, if $f(n)$ is asymptotically linear in $n$, the probability converges exponentially fast to 1.Comment: 11 page

Regular hypermaps over projective linear groups, submitted for publication

An enumeration result for orientably-regular hypermaps of a given type with automorphism groups isomorphic to PSL(2,q) or PGL(2,q) can be extracted from a 1969 paper by Sah. We extend the investigation to orientable reflexible hypermaps and to non-orientable regular hypermaps, providing many more details about the associated computations and explicit generating sets for the associated groups.

Regular maps with almost Sylow-cyclic automorphism groups, and classification of regular maps with Euler characteristic -p²

A regular map M is a cellular decomposition of a surface such that its automorphism group Aut(M) acts transitively on the flags of M. It can be shown that if a Sylow subgroup P ≤ Aut(M) has order coprime to the Euler characteristic of the supporting surface, then P is cyclic or dihedral. This observation motivates the topic of the current paper, where we study regular maps whose automorphism groups have the property that all their Sylow subgroups contain a cyclic subgroup of index at most 2. The main result of the paper is a complete classification of such maps. As an application, we show that no regular maps of Euler characteristic −p² exist for p a prime greater than 7

The genera, reflexibility and simplicity of regular maps

This paper uses combinatorial group theory to help answer some longstanding questions about the genera of orientable surfaces that carry particular kinds of regular maps. By classifying all orientably-regular maps whose automorphism group has order coprime to g − 1, where g is the genus, all orientably-regular maps of genus p + 1 for p prime are determined. As a consequence, it is shown that orientable surfaces of infinitely many genera carry no regular map that is chiral (irreflexible), and that orientable surfaces of infinitely many genera carry no reflexible regular map with simple underlying graph. Another consequence is a simpler proof of the Breda-Nedela- ˇ Siráň classification of non-orientable regular maps of Euler characteristic −p where p is prime

Regular Clique Covers of Graphs

A family of cliques in a graph G is said to be p-regular if any two cliques in the family intersect in exactly p vertices. A graph G is said to have a p-regular k-clique cover if there is a p-regular family H of k-cliques of G such that each edge of G belongs to a clique in H. Such a p-regular k-clique cover is separable if the complete subgraphs of order p that arise as intersections of pairs of distinct cliques of H are mutually vertex-disjoint. For any given integers p; k; `; p! k, we present bounds on the smallest order of a graph that has a p-regular k-clique cover with exactly ` cliques, and we describe all graphs that have p-regular separable k-clique covers with ` cliques