Combinatorics & Algebra Seminar-On the bi-Cayley isomorphism problem
Wednesday, October 29, 2025
4:00 pm - 5:00 pm
Title: On the bi-Cayley isomorphism problem
Speaker: Greg Robson, U. Primorska
Date and time: Wednesday 10/29, 4–5pm
Place: Phillips 108
Speaker: Greg Robson, U. Primorska
Date and time: Wednesday 10/29, 4–5pm
Place: Phillips 108
Abstract: Graphs are point-line incidence structures, often used in mathematics to help visualize algebraic structures by representing group elements as the points and drawing edges according to a chosen incidence rule. Perhaps the most well-known example of such an encoded graph is a Cayley graph, which encodes how elements of a group are related to one another by a fixed set of generators. A Haar graph is a related construction that generalizes directed Cayley graphs (Cayley digraphs) by forcing bipartiteness.
The BCI problem asks: given two Haar graphs built from the same group G, can we tell whether they are the same (i.e. isomorphic) just by looking at a certain list of "obvious" symmetries inherited from G itself?
In this talk, I will describe a broader version of this question, where our "certain list of symmetries" includes one additional map. We will then focus on the case when the group G is abelian. Except for one special situation, we can reduce the BCI problem to a simpler problem about a related quotient or component graph and, in almost all cases, reduce further to the more familiar Cayley isomorphism problem.