CanaDAM 2021
On-line, May 25 - 28, 2021

Movement and symmetry in graphs - Part II
Org: Karen Gunderson (University of Manitoba), Karen Meagher (University of Regina) and Joy Morris (University of Lethbridge)

EDWARD DOBSON, University of Primorska
Recognizing vertex-transitive digraphs which are wreath products and double coset digraphs  [PDF]

We show that a Cayley digraph of a group $G$ with connection set $S$ is isomorphic to a nontrivial wreath product of digraphs if and only if there is a proper nontrivial subgroup $H\le G$ such that $S\setminus H$ is a union of double cosets of $H$ in $G$. We then give applications of this result which include showing the problem of determining automorphism groups of vertex-transitive digraphs is equivalent to the problem of determining automorphism groups of Cayley digraphs.

MAHSA NASROLLAHI, University of Regina
On a generalization of the Erdos-Ko-Rado theorem to intersecting and set-wise intersecting perfect matchings  [PDF]

A perfect matching ($\mathcal{PM}$) in the complete graph $K_{2k}$ is a set of edges in which every vertex is covered exactly once. Two $\mathcal{PM}$s are $t$-intersecting if they have at least $t$ edges in common. Two $\mathcal{PM}$s $P$ and $Q$ of a graph on $2k$ vertices are said to be set-wise $t$-intersecting if there exist edges $P_{1}, \ldots, P_{t}$ in $P$ and $Q_{1}, \ldots, Q_{t}$ in $Q$ whose unions of edges have the same set of vertices. In this talk we show an extension of the famous Erd\H{o}s-Ko-Rado theorem to intersecting and set-wise intersecting $\mathcal{PM}$ for $t=2$ and $t=3$.

VENKATA RAGHU TEJ PANTANGI, Southern University of Science and Technology
Intersecting sets in Permutation groups.  [PDF]

An intersecting set in a transitive permutation group $G \leq Sym(\Omega)$ is a subset $\mathcal{F} \subset G$ such that given $g,h\in \mathcal{F}$, there exists $\omega \in \Omega$ with $\omega^{g}=\omega^{h}$. Cosets of point stabilizers are natural examples of intersecting sets. In view of the classical Erdos-Ko-Rado theorem, it is of interest to find the size of the largest intersecting set. A group is said to satisfy the EKR property if $|\mathcal{F}|\leq|G_{\omega}|$, for every intersecting set $\mathcal{F}$. It is known that $2$-transitive groups satisfy the EKR property. We will show that general permutation groups are ``quite far'' from satisfying the EKR property.

JASON SEMERARO, University of Leicester
Higher tournaments, hypergraphs, automorphisms and extremal results  [PDF]

In 2017, Karen Gunderson and I use switching classes of tournaments to provide constructions of $r$-hypergraphs with the maximum number of hyperedges, subject to the condition that every set of $r+1$ vertices spans at most $2$ hyperedges. Here we assume $r \ge 3$. A $d$-tournament is a set together with an inductively defined orientation on each of its $d$-sets. Generalising results of Babai--Cameron, we show that $3$-tournaments admit a switching operation and use our results to obtain some new lower bounds for extremal numbers.

GABRIEL VERRET, University of Auckland
Regular Cayley maps and skew morphisms of monolithic groups  [PDF]

Skew morphisms, which generalise automorphisms for groups, provide a fundamental tool for the study of regular Cayley maps and, more generally, for finite groups with a complementary factorisation $G = BY$, where $Y$ is cyclic and core-free in $G$. We will explain the connection between these topics and discuss some recent results on the case when $B$ is a monolithic group.