Prairie Mathematics Colloquium

Location: Live Stream with Meet & Greet starting at 2:00 p.m.

Speaker: Shonda Dueck, Associate Professor, University of Winnipeg

Title: Quillen Cyclic Partitions of Complete and Almost Complete Uniform Hypergraphs

Zoom Link:  https://umanitoba.zoom.us/j/64182057866?pwd=n3y6W8jeDVaEs4xASIcz0oQCxZAOv5.1

Meet and Greet: 2:00pm-2:30pm CST
Talk: 2:30pm-3:20pm CST
Questions: 3:20pm-3:30pm CST

Abstract:

 We consider cyclic partitions of the complete \(k\)-uniform hypergraph on a finite set \(V\), minus a set of \(s\) edges, \(s \geq 0\). An \(s\)-almost \(t\)-complementary \(k\)-hypergraph is a \(k\)-uniform hypergraph 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 \(k\)-subsets of \(V\) minus a set of \(s\) edges. Such a permutation \(\theta\) is called an \(s\)-almost \((t,k)\)-complementing permutation. The \(s\)-almost \(t\)-complementary \(k\)-hypergraphs are a natural generalization of the almost self-complementary graphs which were previously studied by Clapham, Kamble et al, and Wojda. We prove the existence of an \(s\)-almost \(p\)-complementary \(k\)-hypergraph of order \(n\), where \(p\) is prime, \(s =\prod_{i \geq 0}\binom{n_i}{k_i}\), and \(n_i\) and \(k_i\) are the entries in the base-\(p\) representations of \(n\) and \(k\), respectively. This existence result yields a combinatorial proof of Lucas' classic 1878 theorem. We also present a construction for vertex-transitive \(q\)-complementary uniform hypergraphs, for any prime power \(q\), which can be viewed as a generalization of the Paley graph construction to uniform hypergraphs.

See:  Prairie Mathematics Colloquium

This event is supported by PIMS.