### Double Bonds and Edge Colorings

There is an effort going on to improve the double bond assignment machinery in the CDK, which is great. Of interest to me, however, is how many possible arrangements of double bonds can you have in fused ring systems. This was mentioned in at least one previous post - or perhaps two.

However, lets get a very rough upper bound; how many ways are there to color the N edges of a graph with two colors? This is 2 to the power N, or the set of all subsets of the edges. Of course, many of these are chemically meaningless, where atoms have too high a valence. So filter out those where adjacent edges have the same color - or more exactly, where adjacent edges are colored with the 'double bond' color (let's call it '2').

The image shows a sketch of the simple procedure (above) and a slightly better approach (below). The better way of doing things is similar to the k-independent chessboard solution (sorry to link to my own pages so much - but it is relevant!). The idea is to use the symmetries of the graph (or chessboard) to prune solutions that must have already been tried.

This does seem to work, but there are a huge number of solutions - even for relatively small graphs. For example, fusanes with just 4 rings have thousands of partial solutions. For these four examples, there is quite a variation:

The most symmetric (green) example has the least - of course - but the large numbers of solutions of size 5 for the orange example is odd. It's a bit difficult to look through these to see why, unfortunately. What is much easier is the 'full' solutions of size 9. For example, for the green graph:

A bit difficult to distinguish these drawings - actual double lines are clearer, it seems. Anyway, below each is a kind of 'name' based on the bond equivalence classes (a-e) in the lower center. So, "b3.d3" means three 'b' bonds and three 'd' bonds are colored.

### 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…

### Generating Dungeons With BSP Trees or Sliceable Rectangles

So, I admit that the original reason for looking at sliceable rectangles was because of this gaming stackoverflow question about generating dungeon maps. The approach described there uses something called a binary split partition tree (BSP Tree) that's usually used in the context of 3D - notably in the rendering engine of the game Doom. Here is a BSP tree, as an example:

In the image, we have a sliced rectangle on the left, with the final rectangles labelled with letters (A-E) and the slices with numbers (1-4). The corresponding tree is on the right, with the slices as internal nodes labelled with 'h' for horizontal and 'v' for vertical. Naturally, only the leaves correspond to rectangles, and each internal node has two children - it's a binary tree.

So what is the connection between such trees and the sliceable dual graphs? Well, the rectangles are related in exactly the expected way:

Here, the same BSP tree is on the left (without some labels), and the slicea…

### Listing Degree Restricted Trees

Although stack overflow is generally just an endless source of questions on the lines of "HALP plz give CODES!? ... NOT homeWORK!! - don't close :(" occasionally you get more interesting ones. For example this one that asks about degree-restricted trees. Also there's some stuff about vertex labelling, but I think I've slightly missed something there.

In any case, lets look at the simpler problem : listing non-isomorphic trees with max degree 3. It's a nice small example of a general approach that I've been thinking about. The idea is to:
Given N vertices, partition 2(N - 1) into N parts of at most 3 -> D = {d0, d1, ... }For each d_i in D, connect the degrees in all possible ways that make trees.Filter out duplicates within each set generated by some d_i. Hmm. Sure would be nice to have maths formatting on blogger....

Anyway, look at this example for partitioning 12 into 7 parts:

At the top are the partitions, in the middle the trees (colored by degree) …