Graph Drawings

Since early 2021, I have had the pleasure of studying De Bruijn sequences, universal partial cycles, and perfect necklaces, all of which can be represented as Hamiltonian or Eulerian circuits on different dimensions of De Bruijn graphs, or the more general astute graphs (also known as Praeger Xu graphs). In fact, my collaborators and I have a submitted paper on this work, available to view on ArXiv here: https://arxiv.org/abs/2310.13067.

Below are some of the hand-drawn (and occasionally LaTeχ-coded) graphs that I have created over the course of my research:

I learned of tri- and hexa-hexaflexagons from Vihart in 2012, and since then I have been fascinated by them, to the extent that I typically keep a hexa-hexaflexagon in my wallet for just such an occasion as presented itself recently when I had the pleasure of introducing a group of combinatorists to the beautiful structure of this novel object and its state diagram. During that discussion, it occurred to me that I hadn't ever tried to construct a higher than six-faced version of this object (I had made many tri- and hexa-hexaflexigons, and even some tetraflexagons). I was very pleased then when my first attempt at construction indeed resulted in a dodeca-hexaflexagon, whose state diagram is most satisfyingly exactly what I would have hoped.