Francesca Merola

Infinitely many cyclic solutions to the Hamilton-Waterloo problem with odd length cycles

It is conjectured that for every pair ( ź , m ) of odd integers greater than 2 with m ź 1 ( mod ź ) , there exists a cyclic two-factorization of K ź m having exactly ( m - 1 ) / 2 factors of type ź m and all the others of type m ź . The authors prove the conjecture in the affirmative when ź ź 1 ( mod 4 ) and m ź ź 2 - ź + 1 .

A collection of results on Hamiltonian cycle systems with a nice automorphism group

Abstract We collect some old and new results on Hamiltonian cycle systems of the complete graph (or the complete graph minus a 1-factor) having an automorphism group that satisfies specific properties.

Product action

This paper studies the cycle indices of products of permutation groups. The main focus is on the product action of the direct product of permutation groups. The number of orbits of the product on n-tuples is trivial to compute from the numbers of orbits of the factors; on the other hand, computing the cycle index of the product is more intricate. Reconciling the two computations leads to some interesting questions about formal power series. We also discuss what happens for infinite (oligomorphic) groups and give detailed examples. Finally, we briefly turn our attention to generalised wreath products, which are a common generalisation of both the direct product with the product action and the wreath product with the imprimitive action.

Parker Vectors for Infinite Groups

We take the first step towards establishing a theory of Parker vectors for infinite permutation groups, with an emphasis towards oligomorphic groups. We show that, on the one hand, many results for finite groups extend naturally to the infinite case (Parker?s Lemma, multiplicative properties, etc.), while on the other, in the infinite case some genuinely new phenomena arise. We also note that calculating Parker vectors of oligomorphic groups is akin to counting circulant combinatorial objects, mirroring in a sense the combinatorial meaning of the orbit-counting sequence of an oligomorphic group. Finally we explicitly find the Parker vectors for some groups, one of which being the automorphism group of the Rado graph.

Orbits onn-tuples for Infinite Permutation Groups

This paper presents a theorem on the growth rate of the orbit-counting sequences of a primitive oligomorphic group: if G is not a highly homogeneous group, then the growth rate for the sequence counting orbits onn -tuples of distinct elements is bounded below by cnn!, wherec? 1.172. The previously known lower bounds concerned all not highly transitive groups, including highly homogeneous groups which are known to have roughly factorial growth rate. This paper shows that highly homogeneous groups are the only groups with such a growth rate, while for all other primitive groups the growth rate is faster and the bound is improved by an exponential factor.

Cyclic and symmetric hamiltonian cycle systems of the complete multipartite graph: even number of parts

In this paper, we present a complete solution to the existence problem for a cyclic hamiltonian cycle system for the complete multipartite graph with an even number of parts all of the same cardinality. We also give necessary and sufficient conditions for the system to be symmetric as well.

Numeration and enumeration

In this paper, numeration systems defined by recurrent sequences are considered. We present a class of recurrences yielding numeration systems for which the words corresponding to greedy expressions for natural numbers are easily described. Those sequences, in turn, enumerate classes of words with forbidden substrings.

Fano Kaleidoscopes and their generalizations

In this work we introduce Fano Kaleidoscopes, Hesse Kaleidoscopes and their generalizations. These are a particular kind of colored designs for which we will discuss general theory, present some constructions and prove existence results. In particular, using difference methods we show the existence of both a Fano and a Hesse Kaleidoscope on v points when v is a prime or prime power congruent to 1

Hamiltonian Cycle Systems Which Are Both Cyclic and Symmetric

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