Edward Dobson

Transitive Permutation Groups of Prime-Squared Degree

AbstractWe explicitly determine all of the transitive groups of degree p2, p a prime, whose Sylow p-subgroup is not isomorphic to the wreath product

The Erdos-Sós conjecture for graphs of girth 5

\mathbb{Z}_p \wr \mathbb{Z}_p

Automorphism groups with cyclic commutator subgroup and Hamilton cycles

. Furthermore, we provide a general description of the transitive groups of degree p2 whose Sylow p-subgroup is isomorphic to

Cayley graphs on abelian groups

\mathbb{Z}_p \wr \mathbb{Z}_p

On groups of odd prime-power degree that contain a full cycle

, and explicitly determine most of them. As applications, we solve the Cayley Isomorphism problem for Cayley objects of an abelian group of order p2, explicitly determine the full automorphism group of Cayley graphs of abelian groups of order p2, and find all nonnormal Cayley graphs of order p2.

Almost self-complementary circulant graphs

Let p and q be distinct primes. We characterize transitive groups G that admit a complete block system of q blocks of size p such that the subgroup of G which fixes each block set-wise has a Sylow p-subgroup of order p. Using this result, we prove that the full automorphism group of a metacirculant graph Γ of order pq such that Aut(Γ) is imprimitive, is contained in one of several families of transitive groups. As the automorphism groups of vertex-transitive graphs of order pq that are primitive have been determined by several authors, this result implies that automorphism groups of vertex-transitive graphs of order pq are known. We also determine all nonnormal Cayley graphs of order pq, and all 1/2-transitive graphs of order pq.