2025 CMS Winter Meeting
Toronto, Dec 5 - 8, 2025
Org: Alice Lacaze-Masmonteil (University of Regina), David Pike (Memorial University of Newfoundland) and Doug Stinson (University of Waterloo)
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- MASOOMEH AKBARI, University of Ottawa
A Complete Solution to the Generalized HOP with One Round Table [PDF]
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The Honeymoon Oberwolfach Problem (HOP), introduced by Šajna, is a recent variant of the classic Oberwolfach Problem. This problem asks whether it is possible to seat $2m_1 + 2m_2 + \dots + 2m_t = 2n$ participants, consisting of $n$ newlywed couples, at $t$ round tables of sizes $2m_1, 2m_2, \dots, 2m_t$ for $2n - 2$ successive nights, so that each participant sits next to their spouse every night and next to every other participant exactly once. This problem is denoted by $\mathrm{HOP}(2m_1, 2m_2, \ldots, 2m_t)$. Jerade, Lepine, and Šajna have studied the HOP and resolved several important cases.
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 present a general approach to this problem and provide a solution to the generalized HOP with one round table, showing that the necessary condition for $\mathrm{HOP}(2^{\langle s \rangle}, 2m)$ to have a solution is also sufficient.
- TIM ALDERSON,, University of New Brunswick- Saint John
- ANDREA BURGESS, University of New Brunswick- Saint John
- AMANDA CHAFEE, Carleton University
- PETER DANZIGER, Toronto Metropolitan University
- SHONDA DUECK, University of Winnipeg
Cyclic partitions of complete hypergraphs and large sets of combinatorial designs [PDF]
- ANDREA BURGESS, University of New Brunswick- Saint John
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We consider cyclic partitions of the complete $k$-uniform hy\-per\-graph on a finite set $V$, and their relationship to combinatorial designs. A $t$-com\-ple\-ment\-ary $k$-hy\-per\-graph is a $k$-uniform hy\-per\-graph with vertex set $V$ and edge set $E$ for which there exists a permutation $\theta\in Sym(V)$ such that the sets $E, E^\theta, E^{\theta^2}, \ldots, E^{\theta^{t-1}}$ partition the set of all $k$-element subsets of $V$. Such a permutation $\theta$ is called a $(t,k)$-com\-ple\-ment\-ing permutation. The $t$-com\-ple\-ment\-ary $k$-hy\-per\-graphs are a natural generalization of the almost self-com\-ple\-ment\-ary graphs, since the associated $(t,k)$-complementing permutation $\theta$ decomposes the complete $k$-uniform hypergraph into $t$ isomorphic $k$-hypergraphs, which are permuted cyclically by $\theta$. When these $t$-complementary $k$-hypergraphs in the decomposition are also regular, then they form a large set of $t$ isomorphic combinatorial designs. We will look at some algebraic constructions for large sets of combinatorial designs that arise from these cyclic decompositions, including one which generalizes the well know Paley graph construction.
- ALENA ERNST, Worcester Polytechnic Institute
- CALEB JONES,, Toronto Metropolitan University
- WILLIAM KELLOUGH, Memorial University of Newfoundland
BIBDs That Almost Have Locally Equitable Colourings [PDF]
- CALEB JONES,, Toronto Metropolitan University
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In this talk, we analyze $\ell$-colourings of $(v,k,\lambda)$-BIBDs where within each block, one colour is absent and the rest appear $\frac{k}{\ell-1}$ times. We give necessary conditions for such colourings to exist. We show how Hadamard matrices, affine planes, and twin prime powers can be used to construct such coloured BIBDs.
- DONALD KREHER, Michigan Technological University
- ALICE LACAZE-MASMONTEIL,, University of Regina
- SUMIN LEEM, University of Calgary
- SHUXING LI, University of Delaware
- TRENT MARBACH, Toronto Metropolitan University
- WILLIAM MARTIN, Worcester Polytechnic Institute
- SHAHRIYAR POURAKBAR SAFFAR, Memorial University of Newfoundland
Constructing uniquely 2-colourable 4-cycle decompositions [PDF]
- ALICE LACAZE-MASMONTEIL,, University of Regina
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A cycle system of order $n$ is a decomposition of the edges of the complete graph $K_n$ into cycles of a fixed length. A cycle system is said to be $k$-colourable if we can assign $k$ colours to its vertices so that no cycle is monochromatic. If a cycle system is $k$-colourable but not $(k-1)$-colourable, it is called $k$-chromatic. A $k$-colourable cycle system is uniquely $k$-colourable if its colouring is unique up to the permutation of colour classes.
The study of colouring cycle systems has been explored in various settings. In particular, Horsley and Pike have examined the existence of $k$-chromatic $m$-cycle systems for any integers $m>2$ and $k>1$. While Forbes has investigated $3$-cycle systems with unique $3$-colourability, the existence of uniquely $k$-colourable $m$-cycle systems in general remains an open problem.
In this talk, we mainly focus on the construction of an infinite family of uniquely $2$-colourable $4$-cycle systems and also a uniquely $2$-colourable $4$-cycle decomposition of $K_n - I$, for infinitely many integers $n \geq 2$. These constructions contribute to the broader study of uniquely colourable cycle systems and open new directions for future research.
- MATEJA SAJNA, University of Ottawa
- KIANOOSH SHOKRI, University of Ottawa
- BRETT STEVENS, Carleton University
- DOUG STINSON,, University of Waterloo
- AMY WIEBE, University of British Columbia, Okanagan
- KIANOOSH SHOKRI, University of Ottawa