We generalize HOP by allowing tables of size two, instead of a minimum size of four as previously defined in HOP. Thus, in the generalized HOP, we are aiming to seat the $2n$ participants at $s$ tables of size $2$ and $t$ round tables of sizes $2m_1, 2m_2, \ldots, 2m_t$ with the assumption that $2n = 2s + 2m_1+2m_2+\ldots+2m_t$. In this talk, we will present a general approach to this problem, and we will show that the generalized HOP has a solution whenever $m_1+m_2+\ldots+m_t\leq 10$, that is, the sum of the table sizes other than size $2$ is at most $20$.

These results can be restated as results on embedding partial symmetric layer-rainbow latin cubes in partial symmetric layer-rainbow latin cubes where all diagonal entries are empty.

Equitable colourings of cycle systems were introduced in work by Adams, Bryant, Lefevre and Waterhouse, who considered equitably $2$- and $3$-colourable $\ell$-cycle systems of $K_v$ and $K_v-I$ for $\ell \in \{4,5,6\}$. In this talk, we discuss some constructions of equitably $2$-colourable $\ell$-cycle decompositions of $K_v$ and $K_v-I$ in the case that $v$ and $\ell$ have the same parity. In particular, we show that there is an equitably $2$-colourable $\ell$-cycle system of $K_v$ whenever $\ell$ is odd and $v \equiv 1$ or $\ell \pmod{2\ell}$.

This is joint work with Francesca Merola.

We determine sufficient conditions for the block intersection graph (BIG) of block size $k$ packings and block size 3 coverings to be Hamiltonian. The BIG of a packing is Hamiltonian for $k$ even if $4[|X||V \backslash X|-\partial(X)] \geq vk$ and for $k$ odd if $4k[|X||V \backslash X|-\partial(X)\geq v(k^2-1)$. The BIG of a covering is Hamiltonian if $v \geq 3$. Because of our interest in DCCD, we are also interested in Hamiltonian cycles in 1-BIG of block size 3 coverings and we discuss our progress in this case.

In this talk we consider colourings of the subclass of Steiner triple systems which are resolvable, namely Kirkman Triple Systems. We show that for each $v\equiv 3\pmod{6}$ there exists a Kirkman Triple System on $v$ points with weak chromatic number $3$. We also show that for each integer $\delta \geq 3$, there exist infinitely many Kirkman triple systems with weak chromatic number $\delta$.

In 1852, Professor Dr. J. Steiner of Berlin, asked for which number $N$ does there exist a set system containing no pairs that has order $N$ and maximum block size $k$ satisfying

(1) no block properly contains another block, and

(2) for all $t=2,3,...,k-1$ every $t$-set that does not contain a block is contained in exactly one block of size $(t+1)$ .

W.H. Bussey from the University of Minnesota in 1914 constructed the only known solution. His construction provided for each $k\geq 5$ a set-system of order $N=2^{k-1}-1$ and maximum block size $k$ that satisfies Steiner's conditions. At the CMS 75th+1 anniversary summer meeting, I presented our investigation on this problem. See:

C.J. Colbourn, D.L. Kreher and P.R.J. Östergård, Bussey systems and Steiner's tactical problem. $Glas.$ $Mat.$ $Ser.$ $III$, web.math.pmf.unizg.hr/glasnik/forthcoming/pGM7100.pdf

Today's discussion will examine what happens when pairs are allowed as blocks. In particular we consider as blocks the edges of the complete multipartite graph $G=K_{n_1,n_2,\ldots,n_r}$ or its complement $\overline{G}$.

In this talk, we consider cover-free families on hypergraphs, which are generalizations of cover-free families used in applications where the possible sets of defective items are specified by the edges of a hypergraph. The traditional group testing problem is the special case where the edges of the hypergraph are all $d$-subsets of an $n$-set. We discuss recent constructions and our ongoing research on this topic.

The primary result we present is a minimum degree threshold for $G_P$ that allows for a fractional tile-decomposition and therefore implies the existence of a fractional completion of $P$. The proof employs spectral decomposition, the properties of coherent configurations, and perturbation theory to estimate a generalized inverse for the matrix representation of a partial Sudoku puzzle in order to find a solution for the relaxed linear system. Improving on this result by finding a minimum degree threshold for an exact tile-decomposition is an interesting open question in this research area.

It is well known that when $M$ is real (having $\pm 1$ entries), then $MM^{\intercal}=nI_n$ implies that $n=1,2$ or $n$ is a multiple of $4$. Therefore, if we consider the maximal determinant of matrices with entries in the set $\{+1,-1\}$, then Hadamard's bound is not achievable at odd orders, or at orders $n\equiv 2\pmod{4}$ larger than $2$. In the literature, one can find improved upper and lower bounds for the determinant of $\pm 1$ matrices — the exact values of the maximal determinant have been determined for small values of n, and through several infinite families of examples. $\ $

In this talk, we consider the more general problem of finding the maximal determinant of matrices with entries taken from the set $\mu_m$ of $m$-th roots of unity, for some fixed value of $m$. We will present new upper and lower bounds for the determinant for a general value of $m$, and study the ternary case $m=3$, and quaternary case $m=4$ in more detail.

The maximality of the determinant sometimes imposes strong regularity conditions that make the matrices be equivalent to certain combinatorial designs. Conversely, constructions making use of designs or finite geometries, often give large values for the determinant. We will pay special attention to such connections.

In the directed version, we are interested in decomposing $K_{n}^\ast$, the complete symmetric digraph of order $n$, into spanning subdigraphs, each a disjoint union of $k$ directed cycles of lengths $m_1, \ldots, m_k$ (where $m_1+ \ldots +m_k=n$). Such a decomposition models a seating arrangement of $n$ participants at $k$ tables of sizes $m_1, \ldots, m_k$ such that everybody sits to the right of everybody else exactly once.

While the Oberwolfach problem for cycles of uniform length was solved decades ago, the solution to the directed version for uniform-length cycles was completed only in 2023, and while many infinite families of cases of the Oberwolfach problem with variable cycle lengths are known to have a solution, very little is known about the directed version with variable cycle lengths. In this talk, we present a recursive construction that generates solutions to many infinite families of cases of the directed Oberwolfach problem with variable cycle lengths. In particular, we obtain an almost-complete solution to the two-table directed Oberwolfach problem.

This is joint work with Suzan Kadri.