### Colorful Expanding Triangulations and Sliceable Rectangular Graphs

There is a whole area of study on visualisations called cartograms - most appealing are the ones that make countries look like inflated or deflated balloons. The rectangular versions of these are less pretty, but more interesting to me from a graph theory perspective.

I came across this subject via an impressive masters thesis by Vincent Kusters : '
Characterizing Graphs with a Sliceable Rectangular Dual' … which is a title that will take some explaining. Firstly, what is a 'rectangular dual' when it's at home? Well check this out:

Clearly the thing on the left is a graph, and on the right is its rectangular dual - in fact, this is the smallest 'sliceable' dual. By sliceable, I mean that the white rectangles can be made by recursively slicing up a rectangle. For example, if a slice is like [{0, 3, 4, 5, 6}, {1, 2}] for making the first split into the areas of 1 and 2 on the right, and all the rest on the left. The next could be [{0}, {3, 4, 5, 6}] and so on.

The colors of the graph indicate a top/bottom cut in red, and a left/right cut in blue. So rectangles 0 and 1 share a left-right (blue) boundary, while 0 and 4 share a top-bottom (red) boundary. The square nodes [T, R, B, L] are the 'corners', and serve to anchor the dual. There's a lot of detail that I'm skipping here, but this is the broad picture.

Interestingly - for me - Kuster's work makes use of a program called Plantri made by none other than Gunnar Brinkmann and Brendan McKay. It generates planar triangulations - which rectangular duals are examples of - and then colors them to make proper duals. The way Plantri works is fairly familiar; canonical path augmentation but with a restricted set of operations to add vertices and edges:

Starting from K4 - the complete graph on 4 vertices - these 'expansions' are applied to graphs while rejecting duplicates using CPA. Now the thing that occurs to me is the possibility of expanding while maintaining the colorings of a rectangular dual. For example:

These are just examples of E5 from the picture before, but starting from particular colorings, and expanding only to particular colorings. As can be seen from the rectangular slices to the side of each graph, these expansions are 'compatible' in some sense with changes in the dual. Whether this is a meaningful operation or not, I'm not sure. There are a number of possible such expansions, but not a huge number. Here are a couple more:

Note that B and C are the same, but expand to different possible colorings. Also that the outer cycle colors are preserved, along with the some of the internal edges. That is no particular coincidence, since they were chosen specifically to preserve as many of the edges colors as possible.

Interesting, but not yet conclusive in any way.

### How many isomers of C4H11N are there?

One of the most popular queries that lands people at this blog is about the isomers of C4H11N - which I suspect may be some kind of organic chemistry question on student homework. In any case, this post will describe how to find all members of a small space like this by hand rather than using software.

Firstly, lets connect all the hydrogens to the heavy atoms (C and N, in this case). For example:

Now eleven hydrogens can be distributed among these five heavy atoms in various ways. In fact this is the problem of partitioning a number into a list of other numbers which I've talked about before. These partitions and (possible) fragment lists are shown here:

One thing to notice is that all partitions have to have 5 parts - even if one of those parts is 0. That's not strictly a partition anymore, but never mind. The other important point is that some of the partitions lead to multiple fragment lists - [3, 3, 2, 2, 1] could have a CH+NH2 or an NH+CH2.

The final step is to connect u…

### The Gale-Ryser Theorem

This is a small aside. While reading a paper by Grüner, Laue, and Meringer on generation by homomorphism they mentioned the Gale-Ryser (GR) theorem. As it turns out, this is a nice small theorem closely related to the better known Erdős-Gallai (EG).

So, GR says that given two partitions of an integer (p and q) there exists a (0, 1) matrixA iff p*dominatesq such that the row sum vector r(A) = p and the column sum vector c(A) = q.

As with most mathematics, that's quite terse and full of terminology like 'dominates' : but it's relatively simple. Here is an example:

The partitions p and q are at the top left, they both sum to 10. Next, p is transposed to get p* = [5, 4, 1] and this is compared to q at the bottom left. Since the sum at each point in the sequence is greater (or equal) for p* than q, the former dominates. One possible matrix is at the top left with the row sum vector to the right, and the column sum vector below.

Finally, the matrix can be interpreted as a bi…

### Havel-Hakimi Algorithm for Generating Graphs from Degree Sequences

A degree sequence is an ordered list of degrees for the vertices of a graph. For example, here are some graphs and their degree sequences:

Clearly, each graph has only one degree sequence, but the reverse is not true - one degree sequence can correspond to many graphs. Finally, an ordered sequence of numbers (d1 >= d2 >= ... >= dn > 0) may not be the degree sequence of a graph - in other words, it is not graphical.

The Havel-Hakimi (HH) theorem gives us a way to test a degree sequence to see if it is graphical or not. As a side-effect, a graph is produced that realises the sequence. Note that it only produces one graph, not all of them. It proceeds by attaching the first vertex of highest degree to the next set of high-degree vertices. If there are none left to attach to, it has either used up all the sequence to produce a graph, or the sequence was not graphical.

The image above shows the HH algorithm at work on the sequence [3, 3, 2, 2, 1, 1]. Unfortunately, this produce…