We generalized the HOP by allowing tables of size two, relaxing the previous restriction that tables must have a minimum size of four. In the generalized HOP, we aim to seat the $2n$ participants at $s$ tables of size $2$ and $t$ round tables of sizes $2m_1, 2m_2, \dots, 2m_t$, where $2n = 2s + 2m_1 + 2m_2 + \dots + 2m_t$ and $m_i \geq 2$, while preserving the adjacency conditions of the HOP. We denote this problem by $\mathrm{HOP}(2^{\langle s \rangle}, 2m_1, \ldots, 2m_t)$.

In this talk, we will present a general approach to this problem, as well as recent solutions to several cases.

Simon R. Blackburn, `Tredoku patterns', https://arxiv.org/html/2407.10752v3.

Expanding on their work, we bound the dimension of partially reduced Latin squares of order $n$ as a vector space using the representation theory of the symmetric group, investigate vanishing fuzzy Latin squares, and compute the fuzzy Latin squares with six terms.

This work has been supervised by Peter Dukes.

This seating arrangement problem translates naturally into a graph theoretic problem. It is equivalent to finding a cycle decomposition of the graph $K_{2n} + (2n-3)I$ (that is, the complete graph $K_{2n}$ with $2n-3$ additional copies of a fixed $1$-factor $I$) into $2$-factors such that each $2$-factor consists of disjoint $I$-alternating cycles of lengths $2m_1, \ldots, 2m_\ell$. So far, solutions to $HOP$ have been established in various cases, including: all $n \leq 20$; the case when all tables are of the same size; the case when all table sizes are divisible by $8$; and the case when $n$ is odd and $OP(m_1,\ldots,m_\ell)$ has a solution.

In this talk, we present a recursive approach to the problem. In particular, we show that if $m_1,\ldots,m_\ell$ are positive even integers, $t$ is an integer satisfying $\sum_{i=1}^\ell m_i < t$, and $HOP(2m_1,\ldots,2m_\ell)$ has a solution, then $HOP(2m_1,\ldots,2m_\ell,2t)$ also has a solution. Moreover, we briefly discuss the case when $m_1,\ldots,m_\ell$ are not all even.

This is joint work with Mateja \v{S}ajna.

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This talk is based on joint work with Chi Hoi Yip (Georgia Institute of Technology).

Mixed-radix reflected and modular Gray codes are natural extensions of the BRGC, where the base of each position is not necessarily binary. Their respective loopless algorithms can be found as Algorithm H and Exercise 77 in Section 7.2.1.1 of TAOCP Vol. 4A by Knuth. In this talk, we discuss that both of these Gray codes built on a mixed-radix base are guided by the same change sequence, which allows them to be generated in parallel. We show that modular Gray codes can be generated using a greedy cyclic increment approach. Furthermore, we present a new family of modular Gray codes starting from any tuple $w$ that can be constructed by using the same greedy approach to generate the next word that is lexicographically greater than or equal to $w$. We show that although this order is not a suffix of the full modular Gray code, its change sequence is a suffix of the latter.

Joint work with Lucia Moura, Brett Stevens, and Aaron Williams.

This is based on joint work with Roghayeh Maleki and Angelot Behajaina.

The study of colouring decompositions has been explored in various settings, but few results are known for unique colourability even for the most prominent families of decompositions such as balanced incomplete block designs and cycle systems. In 2003, Forbes showed that there exist uniquely 3-colourable Steiner triple systems of order $n$ for all admissible $n\geq 25$. More recently, Darijani and Pike proved the existence of uniquely $k$-colourable $e$-star systems (for all $k\geq 2$ and $e\geq 3$) and uniquely 2-colourable $P_4$ systems. Lastly, Burgess, Pike, and Pourakbar-Saffar constructed a uniquely 2-colourable $4$-cycle system of order $n$ for all admissible $n\geq 49$.

In this talk, we mainly focus on the construction of uniquely $k$-colourable $(r,r)$-biclique decompositions of complete multipartite graphs for all $r\geq 2$ and $k\geq 2$ except $(r,k)=(2,2)$. These constructions are our stepping stone to find uniquely $k$-colourable $2r$-cycle systems.

This is joint work with E.S. Mahmoodian.

Such optimal equidistant codes have been studied at least since the 1970's. However, many results have been reproved multiple times, and some early work seems to be not well well known. Also, the most comprehensive solution to the problem seems to be incomplete. In this talk, we review the history of this problem and attempt to give a complete unified treatment of the known results.