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.

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 Hamilton-Waterloo Problem and its Generalizations  [PDF]

A $\{C_m^r,C_n^s \}$-factorization asks for a 2-factorization of $K_v$ (or $K_v-I$), where $r$ of the 2-factors consists of $m$-cycles and $s$ of the 2-factors consists of $n$-cycles. This is the Hamilton-Waterloo Problem(the HWP) with uniform cycle sizes $m$ and $n$. The HWP is an extension of the Oberwolfach problem which asks for isomorphic 2-factors. 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 non-isomorphic 2-factors 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 $2n-2$ 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 $K_{2n}+(2n-3)I$ into 2-factors, each consisting of disjoint $I$-alternating cycles of lengths $2m_1,\ldots,2m_t$. It is also equivalent to a semi-uniform 1-factorization 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.