
Euler characteristics of nonmanifold polycubes
From a block of cubes, remove two nonadjacent and nonopposite cubes. The resulting polycube has a boundary that is not a manifold: between the two removed cubes, there is an edge shared by four squares, but a twodimensional manifold can only have two faces per edge. Nevertheless, we can compute its Euler characteristic as the number of vertices () minus the number of edges () plus the number of square faces (). , the same number we would expect for the Euler characteristic of a topological sphere! What does it mean?

Linkage
 “You know how the \hat command in LaTeΧ puts a caret above a letter? … Well I was thinking it would be funny if someone made a package that made the \hat command put a picture of an actual hat on the symbol instead?” And then Matthew Scroggs and Adam Townsend went ahead and did it ().

Monochromatic grids in colored grids
Color the points of an grid with two colors. How big a monochromatic gridlike subset can you find? By “gridlike” I mean that it should be possible to place equally many horizontal and vertical lines, partitioning the plane into cells each of which contains a single point.

Coloring kinggraphs
Draw a collection of quadrilaterals in the plane, meeting edge to edge, so that they don’t surround any open space (the region they cover is a topological disk) and every vertex interior to the disk touches at least four quadrilaterals. Is it always possible to color the corners of the quadrilaterals with four colors so that all four colors appear in each quadrilateral?

Photos from Barbados
I spent this year’s spring break at Erik Demaine’s annual computational geometry workshop in Barbados again. A few photos of workshop participants:

Linkage
 A 3regular matchstick graph of girth 5 consisting of 54 vertices, Mike Winkler, Peter Dinkelacker, and Stefan Vogel (). The previous smallestknown graph with these properties had 180 vertices, but this one might still not be optimal, as the known lower bound is only 30. I found it difficult to understand the connectivity of the graph from its matchstick representation so I made another drawing of the same graph in a different style:

Linkage
It’s the last day of classes for the winter quarter here at UCI, and a good time for some spring cleaning of old bookmarked links. Probably also a good time for a reminder that Google+ is shutting down in two weeks so if, like me, you still have links to it then you don’t have long to replace them with archived copies before it gets significantly more difficult.

Planar graphs needing many lines
As I said in my previous post my two SoCG papers both involved trying and failing to prove something else, and writing down what I could prove instead. For the one that appears today, “Cubic planar graphs that cannot be drawn on few lines” (arXiv:1903.05256), that something else was Open Problem 16.14 of my book, which can be rephrased as: does there exist a family of planar graphs that cannot be drawn planarly with all vertices on a constant number of convex curves?

Counting polygon triangulations is hard
In some sense both of my accepted papers at SoCG are about a situation where I really wanted to prove something else, wasn’t able to, and wrote up what I could prove instead. The one whose preprint appears today, “Counting polygon triangulations is hard” (arXiv:1903.04737), proves that it’s complete to count the triangulations of a polygon with holes.

Linkage
This seems like a good time to throw in a word of appreciation for archive.org and their wayback machine for making it so easy to make permanent links to online resources that might otherwise go away, such as other people’s Google+ posts. Just search for the link on archive.org and, if it’s not archived already, it will give you a convenient link to immediately archive it. There’s one of those hiding in the links below, and more among the older links on my blog.
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