I suggest to minimize visual clutter, because any new player's brain will be overloaded trying to figure out a board, and less is sometimes more.
I'd remove the forward direction triangles where they can be unambiguously inferred from the baseline. A lower grain contrast wood texture might be a good idea too.
You mentioned you sometimes use five tile colours but you can probably improve it: yes, the four colour theorem guarantees that :)
Displaying them on some tiles only is something I have not thought of. My first reaction is that it may feel harder to read, as tiles with an arrow may look like they would act differently than those without? We may induce the idea that it has a different purpose. So for now, I'd prefer a solution that would look the same on all tiles, I think that's much easier to parse. But then I should probably give this a try before rejecting it.
You do have a good point regarding the wood texture, I've tried other textures, but couldn't find a better one yet. But I'm sure that's possible. Nevertheless for now, you can also display different skins: https://imgur.com/a/zitKlpz
And thanks a lot for mentioning the four colour theorem. I'm looking at its wikipedia page now. I'm sure I came across it many times before, but I did not think about it when tackling this part of the problem. My implementation is here is anyone can spot my mistake: https://github.com/polygonjs/polygonjs/blob/master/src/engin...
And I also realise that, contrary to what I said above, I do not set the colors on the left and right sides. So that will be an easy modification.
After that, I do go breadth first. But I do so by looking at all neighbours of the current tile, simply by taking them in the order given by my internal graph. Instead I should just take those that are connected to another tile that has been already visited.
So thank you for those advice, that's super helpful, I really appreciate. I had to look up "non-surveyable" as well, I'm learning a lot here. That said, I'm having a hard time understanding "as are forced (up to isomorphism)", is that something you could clarify?
Since the "swap green and purple" operation is bijective (a one-to-one, reversible mapping), and preserves all the properties we care about, we can call it an isomorphism. If there's an isomorphism between two solutions, we say that those solutions are isomorphic. Swapping all the greens and purples of any isolated region of the graph (i.e., one surrounded by a strip of black and white) is an isomorphism, if the only property we care about is "is this a colouring of the original graph?".
I was saying it's pointless to search isomorphic solutions if you can help it. If you have isolated regions of the graph yet to be filled, you can halve the time it takes to explore each region's possible colourings.
However, if you consider isolated regions completely separately, you can solve them one after the other (and backtrack once you know any region's been made impossible by the choices of colours in its border), for a much larger speedup.
The whole thing about isomorphisms was a red herring, an artefact of my thought process rather than an actual insight. I doubt trying to exploit this will net you anything more than a 2× improvement, and that's only if you do it cleverly.
Although I'll have to re-read you when I rework the implementation. But in any case, the operation is already done very quickly, so I've never thought about monitoring it. So any performance improvement on it isn't a high priority. There are other operations in the full grid generation that are more time consuming, like the smoothing/squarification, which needs to be applied multiple times. But even then, since the grid is so small, the whole thing takes less than a second.
That said, I'm thinking of ways to improve the menu where players select the next board. Right now, when you want to generate a random one, I only display one. But it could be nice to display multiple ones. In which case 1 second generation becomes X seconds. So if I do that, any speed up could be welcome.