User leah wrenn berman - MathOverflow

Coloring toroidal polyhedra with convex faces?

2011-02-03T19:26:24Z

Consider a toroidal polyhedron, which is a topological torus, in which all faces are planar, two faces meet in at most an edge, and adjacent faces are not coplanar. The Szilassi polyhedron has 7 non-convex hexagonal faces meeting 3 at a vertex, and every face meets every other face, so it requires 7 colors.

In 1982 ("Coloring Polyhedral Manifolds"), Barnette proved if you require that every face of the toroidal polyhedron be convex, then the polyhedron can be colored in at most 6 colors, and remarked that all the examples that he knew about could be colored in only 4 colors.

(1) Are there known examples which require 5 colors? 6 colors? Alternately, are there any more recent results on colorability of toroidal polyhedra with convex faces?

(2) Are there any known results about coloring toroidal polyhedra embedded on n-holed tori? Is there a "Szilassi polyhedron analogue" for larger genus? Are there known results if you impose convexity?

Answer by Leah Wrenn Berman for Good combinatorics textbooks for teaching undergraduates?

2010-06-28T21:37:21Z

I liked Roberts & Tesman, Applied Combinatorics (2nd edition), but it's out of print. It has nice applications and nice references.

Answer by Leah Wrenn Berman for Sources for Bibtex entries

2010-04-07T21:14:07Z

If you use a mac, BibDesk is fantastic: among other really nice features, it lets you find your book/article/etc on your choice of free sites (ACM, arXiv, CiteULike, Google Scholar, HubMed, SPIRES) or subscription sites (IEEE Xplore, MathSciNet, Project Euclid, Zentralblatt Math) and then once you've found the item, it takes one click to import the citation into the database. The database can also store electronic copies of articles (if available) referenced to the citation.

So for example, I would open BibDesk, click on the icon that says Web, click on "MathSciNet". Within the program I see the MathSciNet page (assuming I'm at work where I have access). Type in the search terms Hartshorne and Geometry, and up comes 8 citations I could import. One of them is Algebraic Geometry from 1977, so I would click on the button that says "import". BibDesk does all the other work.

While writing a paper, just drag and drop the citations onto your LaTeX document to embed \cite{blah} with the appropriate cite key (at least if you're using TeXShop).

When you're ready to stick a bibliography into your paper, select the relevant articles in the BibDesk database and export them into a BibTeX file. It's super easy.

Answer by Leah Wrenn Berman for Is this an if-and-only-if definition of affine?

2010-04-03T17:01:24Z

There seems to be some disagreement about what precisely the definition of an affine transformation is. For example, Martin, in his transformation geometry book, proves that an affine transformation is a map which takes any three noncollinear points to any other three noncollinear points (this is sometimes called the fundamental theorem of affine geometry). In particular, this means that every affine transformation may be represented as the composition of an invertible linear transformation and a translation.

On the other hand, some references simply define an affine transformation as the composition of a linear transformation and a translation, in which case the transformation need not be invertible.

(I'm not familiar with the second reference; it was just handy on a Google Books search on Affine transformations.)

Answer by Leah Wrenn Berman for Math paper authors' order

2010-04-01T23:31:14Z

I had always heard that there was a famous counterexample to alphabetization, the Zucker-Cox Theorem (where they flipped the order for obvious reasons), but apparently the non-alphebetization in this case was apocryphal.

But indeed, non-alphabetization is very rare.

Answer by Leah Wrenn Berman for The Symmetry of a Soccer Ball

2010-03-30T19:22:45Z

If you are willing to relax the trivalency requirement (and not require convexity, which was not one of the stated constraints), you can make all sorts of polyhedra, using only regular pentagons and hexagons, which have all sorts of symmetry.

For example, start with two truncated icosahedra, and remove one hexagon from each of them. Now you can glue them together along the removed hexagons (matching up pentagonal and hexagonal faces), forming a new polyhedron with 3 and 4-valent vertices and considerably less symmetry than what you started with. Continuing, you could make long rod-shaped polyhedra, or you could make some spiky polyhedron by replacing multiple hexagons with other truncated icosahedra.

Answer by Leah Wrenn Berman for Approximate solutions for trisecting the angle or squaring the circle

2010-03-28T01:31:32Z

Do you know about the (Archimedean) solution to trisecting the angle if you allow a marked ruler?

Are combinatorial configurations whose Levi graphs may be represented as covering graphs over voltage graphs realizable with pseudolines?

2010-03-19T22:04:37Z

This question is related to this previous question. Many combinatorial configurations have Levi graphs which may be represented as derived graphs obtained from voltage graphs over a cyclic group; in a number of such cases, it is possible to represent the combinatorial configuration as a geometric configuration (i.e., using points and straight lines in the Euclidean plane).

Given a bipartite graph which is obtained from a voltage graph, we can view it as a Levi graph of some combinatorial configuration. Is it possible to draw all such configurations using pseudolines? If not, are there easy/known constraints on the ones that fail? (e.g., if there are more than x points in the configuration, then things work? You can't use such-and-so groups as the cyclic group for the voltage graph?)

(Does the Heawood graph have a voltage-graph representation? If so, it makes the first question easy to answer, but the second one is still interesting. Maybe.)

Answer by Leah Wrenn Berman for Books you would like to see translated into English.

2010-03-11T00:43:51Z

Friedrich Levi, Geometrische Konfigurationen

Looking for cubic, bipartite graphs with girth at least six and no cycles of length 8.

2009-10-18T00:31:28Z

Aside from the Desargues graph, are there nice (at least vertex-transitive), small (say, less than 60 vertices), cubic, bipartite graphs with girth at least 6 and no 8-cycles? (or, even better, no cycles of length congruent to 0 mod 4.)

Bonus: any idea how to use Mathematica to pull these out of its list of graphs it's got data for in Mathematica 7?

Comment by Leah Wrenn Berman

2010-03-20T18:10:43Z

Would Mnev's Universality Theorem address the realizability question with straight lines? My (basically nonexistent) understanding was that it would/could apply via oriented matroids, and thus would have something to say about realizations by pseudolines. (I guess you could then ask if the pseudoline configuration was stretchable, but that's hard too...)

Comment by Leah Wrenn Berman

2010-03-15T20:10:25Z

I was going to say Grunbaum & Shephard, Tilings and Patterns, but it turns out it's being re-issued---in paperback, no less---later this year. I love dover books.

Comment by Leah Wrenn Berman

2010-03-12T04:56:37Z

I don't really think of the dynamic surveys at EJC to be the kind of expository paper you're talking about, though---while all (most?) surveys are expository, not all expository articles are surveys. (And some of those surveys hardly count as articles: 219 pages (the survey on Graph Labelling) is a lot!)

Comment by Leah Wrenn Berman

2010-03-12T03:01:23Z

Gr\"{u}nbaum's book definitely does not give an answer.