Computational Combinatorics

The Erdős Discrepancy Problem

Computational combinatorics made a big splash last spring when Konev and Lisista announced their SAT-solver-based proof of the Erdős Discrepancy Problem for constant C=2. Recently, Le Bras, Gomes, and Selman announced another SAT-based solution for C=3, but when restricting the sequences to have a special property.

I was working on a draft blog post to talk about these results, but then Terry Tao went and solved the full conjecture. I tweeted about it and got many responses, my favorite being simply the following image:


To help those who have difficulty with Tao’s paper (and I am not qualified to read any part of his solution), I made a short video describing the problem, or read below.

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Hypohamiltonian Planar Graphs

Brendan McKay uploaded a short-but-sweet article to the arXiv titled Hypohamiltonian planar cubic graphs with girth five. Let’s quickly break down all of the words of this title:

  1. A graph is hypohamiltonian if it does not contain a hamilton cycle, but the deletion of any vertex does create a hamilton cycle.
  2. A planar graph is a graph that can be embedded in the plane with no edge crossings.
  3. A cubic graph is a 3-regular graph.
  4. The girth of a graph is the length of the shortest cycle.

McKay details the search for the smallest (and first known) graphs of this type, which have order 76. There are three non-isomorphic examples. The data for these graphs (and more) is available online [Note: at this time, it appears the data is not yet online].

The three smallest examples of hypohamiltonian planar cubic graphs of girth 5. The numbers give the face lengths for faces other than 5-faces. Figures by B.D. McKay.

The three smallest examples of hypohamiltonian planar cubic graphs of girth 5. The numbers give the face lengths for faces other than 5-faces. Figures by B.D. McKay.

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Happy Birthday, Joan Hutchinson!

Bernard Lidický and I organized an AMS Special Session on Extremal and Structural Graph Theory last weekend. It was a wonderful session, filled with wonderful speakers talking about some amazing mathematics.

Notably missing was Joan Hutchinson, who we invited but she declined because she was celebrating a milestone birthday on Sunday. That’s a very good reason to miss the session, and we missed her greatly. However, we did want to share our well-wishes.

Thanks to all who helped with the session and the making of this video! A few participants were missing at the time of filming, but you can try finding your favorite graph theorists in the (very short) video above.

Shannon Capacity of Odd Cycles

Breaking news: Mathew and Östergård have improved lower bounds on the Shannon capacity of some odd cycles. These improvements over a very old and difficult problem come via some sophisticated algorithms.
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Announcement: Computational Combinatorics at CanaDAM

A quick announcement to those who may be interested. Jan Goedgebeur, who has done many nice things including the wonderful House of Graphs, is organizing a Minisymposium at CanaDAM this summer. The minisymposium is Algorithmic Construction of Combinatorial Objects and will be on June 2, after Brendan McKay’s plenary talk (regular readers will remember him from several topics). See the CanaDAM 2015 page for more details about the larger conference. Unfortunately, I cannot attend, as I will be busy at that time.

GRWC 2015 Announcement

We take a small break from our usual discussions in order to announce that applications for GRWC 2015 are now online! See the announcement below.

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On the arXiv

Whenever I write a paper, I put it on the arXiv. The arXiv is an open-access paper repository run by Cornell University. It’s pretty fantastic to know that almost anyone can have an immediate international audience by posting a paper to the arXiv. My use is two-fold: I upload papers and I look forward to the evening paper dump in my RSS feed five times a week. It is a great way to be actively connected to the research world. As such, I try to convince my coauthors to upload papers to the arXiv and have never had one complaint.

That is, until something interesting happened. I’ll tell the story very quickly, then talk about pros and cons for using the arXiv. This will entirely skip any copyright issues with journals, and focus on the benefits of public/private preprints. Please add your comments!
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Boron and Buckyballs

Recent news from the world of chemistry: the result is really the experimental observation of a special new molecule. People call it a “Boron Buckyball”, but this irritates me since we know that the buckyball is a specific fullerene on 60 vertices. In particular, it is the smallest fullerene that satisfies the independent pentagon rule. This new Boron molecule is called “Borospherene” and is quite interesting in its own right, as seen below:

The borospherene molecule. Credit: Zhai et al.

The borospherene molecule. Credit: Zhai et al.

The structure is very interesting when you consider it as a spherically-embedded (i.e. planar) graph: there are many triangular faces which create a cube-like structure, two of the faces of this cube are 6-faces, and the other four are 7-faces! These interesting heptagonal structures are particularly interesting. In the figure above, it appears to be a unit-distance embedding, and this creates a rotation in the opposing 6-faces.

Let’s dig more into the structure of this object, but also into the computational part of the experiment.
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On Reproducibility

Recently, Stephen Hartke spoke about our work on uniquely K_r-saturated graphs, and afterwards he spoke to a very famous mathematician, who said, roughly:

“Why did you talk about the computation? You should just talk about the results and the proofs and hide the fact that you used computation.”

While this cannot be an exact quote, I believe it is a common attitude among mathematicians. While the focus is on solving problems, the use of computation is seen as a negative, so the final research papers make little mention of their computational methods. This is a huge problem in my opinion. While computations can report exact results, and sometimes even prove results, every execution is an experiment. It is unknown before the execution what the results will be, or how long it will take.

Computational combinatorics is a combination of mathematics, engineering, and science: We prove things, we build things, and we experiment. Since computer proofs are experiments, it is important that they be reproducible. Today, I want to discuss a bit about how we can improve our presentation of algorithms and computation in order to make results more reproducible.
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Rainbow Arithmetic Progressions II: The Collaboration

In the previous post I briefly discussed the algorithms that played a role in the recent paper Rainbow arithmetic progressions. Today, I want to take a detour from our typical discussion of computational methods and instead discuss the collaboration that led to this paper.

It is important to explicitly say that while this paper was very collaborative, and used the expertise and hard work of many individuals, this blog post was written entirely by me during office hours and in a hurry. Any opinions, errors, or other reason to be angry with the content here is entirely my fault and not the fault of my fantastic coauthors. While I’m at it, let me actually list my coauthors by name:

Steve Butler, Craig Erickson, Leslie Hogben, Kirsten Hogenson, Lucas Kramer, Richard L. Kramer, Jephian C.-H. Lin, Ryan R. Martin, Nathan Warnberg, and Michael Young.
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