Design Theory  Part II
Org:
Andrea Burgess (University of New Brunswick),
Peter Danziger (Ryerson University) and
David Pike (Memorial University of Newfoundland)
[
PDF]
 MARCO BURATTI, Università degli Studi di Perugia
Cyclic designs: some selected topics [PDF]

I will choose a couple of topics related to some recent progress on the construction of {\it cyclic} combinatorial designs.
 SAAD ELZANATI, Illinois State University
On edge orbits and hypergraph designs [PDF]

Let $\mathbb{Z}_n$ denote the group of integers modulo $n$ and
let $\mathcal{E}^{(k)}_n$ be the set of all $k$element subsets of $\mathbb{Z}_n$ where $1 \leq k < n$.
If $E \in \mathcal{E}^{(k)}_n$, let $[E]=\{E+r : r \in\mathbb{Z}_n\}$.
Then $[E]$ is the orbit of $E$ where $\mathbb{Z}_n$ acts on $\mathcal{E}^{(k)}_n$ via $(r,E)\mapsto E+r$.
Furthermore, $\{[E] : E\in \mathcal{E}^{(k)}_n\}$ is a partition of $\mathcal{E}^{(k)}_n$ into $\mathbb{Z}_n$orbits.
We count the number of $\mathbb{Z}_n$orbits in $\mathcal{E}^{(k)}_n$ and give the corresponding results when fixed points are introduced. We also give applications to cyclic and $r$pyramidal decompositions of certain classes of uniform hypergraphs into isomorphic subgraphs.
 FRANCESCA MEROLA, Università Roma Tre
Cycle systems of the complete multipartite graph [PDF]

An $\ell$cycle system of a graph $\Gamma$ is a set of $\ell$cycles of $\Gamma$
whose edges partition the edge set of $\Gamma$; is it regular if there is an automorphism group $G$ of $\Gamma$ acting regularly on the vertices of $\Gamma$ and permuting the cycles, cyclic if $G$ is cyclic.
When $\Gamma$ is $K_{m}[n]$, the complete multipartite graph with $m$ parts each of size $n$, the existence problem for $\ell$cycle systems is not completely solved, and little is known on regular systems. We discuss new existence results for cycle systems of $K_{m}[n]$, concentrating mostly on cyclic systems.
 SIBEL OZKAN, Gebze Technical University
On The HamiltonWaterloo Problem and its Generalizations [PDF]

A $\{C_m^r,C_n^s \}$factorization asks for a 2factorization of $K_v$ (or $K_vI$), where $r$ of the 2factors consists of $m$cycles and $s$ of the 2factors consists of $n$cycles. This is the HamiltonWaterloo Problem(the HWP) with uniform cycle sizes $m$ and $n$. The HWP is an extension of the Oberwolfach problem which asks for isomorphic 2factors. We will focus on the HWP with uniform cycle sizes; results on the various lengths of cycles as well as some generalizations to multipartite graphs and also having more nonisomorphic 2factors will be presented.
Results are from the joint works with Keranen, Odabasi, and Ozbay.
 MATEJA SAJNA, University of Ottawa
On the Honeymoon Oberwolfach Problem [PDF]

The Honeymoon Oberwolfach Problem HOP$(2m_1,\ldots,2m_t)$ asks whether is it possible to arrange $n=m_1+\ldots+m_t$ couples at a conference at $t$ round tables of sizes $2m_1,\ldots,2m_t$ for $2n2$ meals so that each participant sits next to their spouse at every meal, and sits next to every other participant exactly once. A solution to HOP$(2m_1,\ldots,2m_t)$ is a decomposition of $K_{2n}+(2n3)I$ into 2factors, each consisting of disjoint $I$alternating cycles of lengths $2m_1,\ldots,2m_t$. It is also equivalent to a semiuniform 1factorization of $K_{2n}$ of type $(2m_1,\ldots,2m_t)$.
We present several results, most notably, a complete solution to the case with uniform cycle lengths.