Improving A Counterexample To Grunbaum’s Conjecture

We left off last time with the possibility of exploring a new family of graphs. Unfortunately, computer computation of edge colorability revealed that it was not a snark. 

However, this week marks an incredible discovery that, if correct, should disprove Grunbaum’s Conjecture for all surfaces of genus at least 3.

Constructing The Counterexample

My work process was heavily motivated by construction performed by Kochol for the case of genus 5. In particular, it involved performing the same “inversion” map as Kochol to attain equivalent presentations. 

I began by attempting to repeat Kochol’s inversion on the Blanusa snark, which has the following presentation on the torus (shown in a paper of Mohar): 

When trying to perform the inversion map, I ran into some trouble trying to visualize the different possible edge connections. In order to resolve this, I even tried constructing a model out of paper to help me visualize the embedding. 

In the end, what I settled on was a general procedure to perform the transformation on any given face as follows. First, we realign until the desired face is in the center. 

Then, treating the desired face as an origin, perform an inversion map ( f(z) = 1/z ), to obtain the final embedding (this is what inspired me to call Kochol’s transformation an “inversion”). Labeling vertices and faces helped greatly with the last step. From now on, we refer to vertices 1 and 6 as x and y respectively. 

To make the graph a less burdensome to draw, we replace with the following representation, calling it B1 for the Blanusa snark.

At this, we follow Kochol’s construction of performing superposition on the graph G26, which was mentioned in an earlier blog. 

This graph has defect 2, so imposing B1 onto edges e1 and e2 will yield a polyhedral embedding of a snark. This graph has genus 3 by construction. 

Moreover, it now follows that the conjecture is also false for genus 4 surfaces, and there are in fact infinitely many counterexamples.

Future Goals

Being able to arrive at my first tangible result has been incredibly rewarding, and I never expected to reach this point. For now, I plan on continuing to pursue this project using other methods, but in the meantime, I have another project I want to tackle. More specifically, I will be examining homology groups of finite topological spaces, coming up with various statistics for different spaces. In particular, the question of finding the minimum number of points to make a space with homology of a sphere is something I hope to answer.

References

1. Kochol, Martin. (2008). Polyhedral Embeddings Of Snarks In Orientable Surfaces. Proceedings Of The American Mathematical Society – PROC AMER MATH SOC. 137. 1613-1619. 10.1090/S0002-9939-08-09698-6.
2. Mohar, B., Steffen, E., & Vodopivec, A. (2008). Relating Embedding And Coloring Properties Of Snarks. Ars Math. Contemp., 1, 169-184.