 As much as I like to pretend that I can do this and just the reality of the most of this is that it's all the four final graph theorists are not particularly clever, but people who do think that graph theory and in particular topologists are. So let me talk about it, it is an outline, a particular kind of ground to colouring define a Hamilton cycle in a ghost room, every one of the vertices exactly once. And so, the motivation part is always difficult. The main thing we should say about it is it's something people have done before and it's kind of rare in what you're at. I have this when the number of vertices is two edges to every vertex, symmetries. So rather than trying to draw a really big diameter, another path going straight through the middle, two edges of every single distance going out. To generalise this to you, I have this first Hamilton cycle, the next one by simply one step. So for example, the next one, this way, I'll rotate this one, and each edge starts to ask if it's cooled on a single edge with a single edge. It should give us a cycle or more generally, if I start off with, this is the reason for this notation, apparently it's using Doug Rust's book. Writing out what this means, if this is the Cartesian product of the vertex centre, then each column is a copy of G and each row is a G. Because this operation seems to do a general thing of taking graphs and turning them into graphs that look like they have Hamilton cycles. This seems like a natural process for building cycles of building graphs that might have... S-H goes in and out saying, oh, this is 0, join 0 and 1 because they're inside the graph structure. And so any algebraic arguments, in particular like the rotations we saw earlier, the decomposition of... This is the kind of scenario in which such arguments are most likely to be able to generalise, so we'd like to ask the same thing. Vertex still has even bigger generators. Same S as a generating set, sort of allows us to be helping the generation, but sort of screw us up. I guess we should stump a writing property of Hamiltonicity to be, it keeps all of the vertices, but I mean, you can't really have it cycling. If I were to draw this graph into, I guess, two of these Hamilton cycles, or two of these double-grains. It's got to touch all of the vertices to the right and these to the left, so it can't be connected. It's got to cross somewhere. But those that there are two edges in one column, this is now a... And then any points that are further off into the column were then not reached by... It does everything okay to the left. Two independent grids trying to... Which says that, okay, assume that it was correct. A double ray of the spiral looks like a one-ended ray. Rather than having this red rectangle into a single larger cycle, so they aren't the 50 pages of case checking to pick up, they're trying to follow the intervals of their ink. The advantage of that is that some more as we can continue making small, finite changes to these infinite double-grains and by some other things, it's such a way that actually these two particular rays end up becoming the same component. So, inside that region are the same component. And if I do this theorem of this form, so each column component will grow as a part, eventually become the entire vertex, eventually...