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Periodic coloring of infinite planar graphs
The de Bruijn–Erdős theorem states that the number of colors needed to color an infinite graph is the same as the maximum number needed for its finite subgraphs. So for any reasonable definition of an infinite planar graph, the 4-color theorem for finite planar graphs implies that every infinite planar graph is also 4-colorable.
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Linkage with two research problems
In case you’ve been worrying that the recent publicity blitz of LLM solutions to open mathematics problems is causing us to run short, there are two more mixed in among my usual links here. Because I haven’t solved them, they are not very precisely formulated, and I don’t know how difficult or interesting they are nor even whether someone else might have already considered them; that’s often the way at the start of research.
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Fractional rings of tangent spheres
Soddy’s hexlet consists of a ring of six spheres, tangent to each other consecutively around the ring, and another ring of three consecutively-tangent spheres, so that all the spheres in the first ring are tangent to all the spheres in the second ring. If you keep one ring fixed, you can rotate the other ring continuously, possibly changing the sizes of some of the spheres as they rotate but keeping the pattern of tangencies unchanged. Here’s a nice animation I found on Wikipedia, where the ring of six spheres rotates continuously while the other ring of three spheres (the central blue one and the two green planes, considered as degenerate spheres tangent at infinity) stays fixed. The larger red sphere is not part of this configuration and I don’t know why the author of this animation included it.
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Linkage
- A permutation generation algorithm in the work of 13th-century Kabbalist Abraham Abulafia (\(\mathbb{M}\), via). The resulting permutation sequence is the one you get by reversing suffixes whose lengths form the sequence \((((2, 3)^2, 2, 4)^3, 2, 5)^4, \dots\) but that’s not the generation rule. Instead the rule is: to generate the permutations of \(1, 2, 3,\dots, n,\) form its \(n\) cyclically rotated permutations (starting with \(1, 2, 3,\dots, n,\)) and for each one in order, recursively generate the permutations of its length-\((n-1)\) suffix.
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Non-coplanar unit distances
Reports that LLMs have killed the Erdős unit distance problem turn out to be greatly exaggerated. There is still plenty not yet understood about the problem.
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Linkage
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Integer complexity and cographs
The integer complexity of a number \(n\) is the minimum number of ones needed to express \(n\) as a parenthesized combination of sums and products of ones. For instance, 10 has complexity 7 as it can be expressed using seven ones, but not fewer:
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Linkage
- Another mathematics journal leaving its commercial publisher (\(\mathbb{M}\)), but with a twist: usually this is accomplished by a mass resignation of the editorial board. But in this case, Communications on Pure and Applied Mathematics is owned by the Courant Institute and was published by Wiley, so taking it in-house is just a matter of not renewing the contract. The causes of friction were increased publisher interference with editorial decisions (the usual), but also editor dissatisfaction with the publisher’s editorial management software.
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Packing Latin squares into sudoku puzzles
I have another new preprint, the result of a research project with UC Irvine undergraduate Cindy Zhang: “Sudoku grids that require many clues” (arXiv:2607.05728, to appear at JCDCG3 2026). The main result is, I think, surprising: When generalized to \(n^2\times n^2\) grids, almost all sudoku puzzles must be almost entirely covered by clues, leaving only a logarithmic fraction of cells blank. This implies an average case time for solving randomly chosen puzzles that is exponential in \(n^4/\log n\), significantly better than the exponential in \(n^4\) that one gets for formulating the problem as an exact cover problem without using this bound on blank cells or the exponential in \(n^4\log n\) that one gets for a brute force search.
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Beachhenge linkage
- On the scale of stupid things the US government is doing this is pretty small, but they appear to be banning the census bureau from any effective methods of privacy-preserving information release, and in particular from adding noise to their data to help create differential privacy (\(\mathbb{M}\), via). Sadly, taking away valuable disclosure avoidance tools doesn’t make fundamental trade-offs go away.
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