Steiner triple systems with transrotational automorphisms

AbstractA Steiner triple system of order v is said to be k-transrotational if it admits an automorphism consisting of a fixed point, a transposition, and k cycles of length (v−3)k. Necessary and sufficient conditions are given for the existence of 1- and 2-transrotational Steiner triple systems

On the full automorphism group of a Hamiltonian cycle system of odd order

It is shown that a necessary condition for an abstract group G to be the full automorphism group of a Hamiltonian cycle system is that G has odd order or it is either binary, or the affine linear group AGL(1; p) with p prime. We show that this condition is also sufficient except possibly for the class of non-solvable binary groups.Comment: 11 pages, 2 figure

The 2-rotational Steiner triple systems of order 25

AbstractIn this paper, we enumerate the 2-rotational Steiner triple systems of order 25. There are exactly 140 pairwise non-isomorphic such designs. All these designs have full automorphism groups of order 12. We also investigate the existence of subsystems and quadrilaterals in these designs

Cyclic, f-Cyclic, and Bicyclic Decompositions of the Complete Graph into the 4-Cycle with a Pendant Edge.

In this paper, we consider decompositions of the complete graph on v vertices into 4-cycles with a pendant edge. In part, we will consider decompositions which admit automorphisms consisting of: (1) a single cycle of length v, (2) f fixed points and a cycle of length v − f, or (3) two disjoint cycles. The purpose of this thesis is to give necessary and sufficient conditions for the existence of cyclic, f-cyclic, and bicyclic Q-decompositions of Kv

On the seven non-isomorphic solutions of the fifteen schoolgirl problem

In this paper we give a simple and effective tool to analyze a given Kirkman triple system of order 15 and determine which of the seven well-known non-isomorphic KTS(15)s it is isomorphic to. Our technique refines and improves the lacing of distinct parallel classes introduced by F. N. Cole, by means of the notion of residual triple defined by G. Falcone and the present author in a previous paper. Unlike Cole's original lacing scheme, our algorithm allows one to distinguish two KTS(15)s also in the harder case where the two systems have the same underlying Steiner triple system. In the special case where the common STS is #19, an alternative method is given by means of the 1-factorizations of the complete graph K_8 associated to the two KTSs. Moreover, we present three new visual solutions to the schoolgirl problem, and we catalogue most of the classical (or interesting) solutions in the literature in terms of what KTS(15)s they are isomorphic to. This paper provides background on a classical topic, while shedding new light on the problem as well

A Method for Classification of Doubly Resolvable Designs and Its Application

This article presents the principal results of the Ph.D. thesis Investigation and classification of doubly resolvable designs by Stela Zhelezova (Institute of Mathematics and Informatics, BAS), successfully defended at the Specialized Academic Council for Informatics and Mathematical Modeling on 22 February 2010.The resolvability of combinatorial designs is intensively investigated because of its applications. This research focuses on resolvable designs with an additional property - they have resolutions which are mutually orthogonal. Such designs are called doubly resolvable. Their specific properties can be used in statistical and cryptographic applications.Therefore the classification of doubly resolvable designs and their sets of mutually orthogonal resolutions might be very important. We develop a method for classification of doubly resolvable designs. Using this method and extending it with some theoretical restrictions we succeed in obtaining a classification of doubly resolvable designs with small parameters. Also we classify 1-parallelisms and 2-parallelisms of PG(5,2) with automorphisms of order 31 and find the first known transitive 2-parallelisms among them. The content of the paper comprises the essentials of the author’s Ph.D. thesis