 Now that I've introduced you to Oilers Walk, let's look at a Hamilton Walk, William Rohn Hamilton. Now, I think the story goes about 1859, there about 1850s he needed some drinking money and he developed this puzzle game, it's just a map with cities and he said could you travel to all the cities, you say you have to visit all the cities but you can only visit every city once, so you can't go past that city again. So, where the Oilers Walk was about the edges, as far as the Hamilton Walks are concerned we are concerned just with visiting all of the nodes, all of the nodes and I can use them just once. Again, we have this concept of an open walk and we have this concept of a closed walk. So, you've got to look at that, so let's look at these two and just have a look. Let's just look at the first graph here, an open walk. Well, we can see as long as, well let's start at one, I can go one, two, three, four, five. I visited all of them and I visited all of them only once, one, two, five, four, three. And I can't go three, two, one, two, five, two, one because as far as getting to one and I haven't done all of the, or if I start here, I can't go back to two but I could do five, four, three, two, one that I could do as well. These are just open ones. But if I wanted to do a closed one, how would I do a closed one here? Is it possible to do a closed one? In other words, I start and I stop at the same, if I go to two, you know, I can go all the way, I've got to get back to one, so that's very difficult. If I go four, three, any other way, five, four, three, two, one, two, I can't come back to two. So, this two seems to be problematic and that I can't get out of there. What if I look at the second one? Can I do a closed Hamilton walk here? Five, four, three, four, five, six, seven, eight, nine, one. So that seems to be, you know, this seems to be, seems difficult when you just look at it but if there's really many, or there's really more than one closed walk you can really take there and I can start at, you know, with it going that way around I can really start anywhere and I can still get to, so I can go nine, one, and I ended nine. So there's many closed walks, there's many closed walks there. So this is, based on this, if you remember this puzzle game, you have a map with cities and you want to visit all the cities, you have to get to all the cities but you can only visit the city once. So you can't come back to that city, you can't visit it twice, just once. So in short that is a Hamilton walk. So yesterday we looked at these Eulerian walks and cycles, let's just today look at the Hamilton walk. So let's create a graph that we had on the board and what I'm going to do is I'm just going to copy and paste it from my other screen on the left-hand side here, let's go there, there we go. And I'm going to hit shift enter, shift return. So slightly different orientation but that's exactly the one we've just had on the board. And I can ask, I can ask whether this is a Hamiltonian graph. Once again remember it's going to look for closed, it's going to find a closed Hamiltonian cycle. So I can just say Hamiltonian graph Q is this a Hamiltonian graph and it's going to say yes it is, let's find a Hamiltonian cycle. A Hamiltonian cycle in G and it's going to find the first one and it says 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 8, 9 and 9 back to 1. And I can actually do a highlight graph of this. Let's have a look at this highlight, a highlight graph of G and what I want there is a path graph, show me the first one, so that's the one that it's found now of this which we just did which was a find Hamiltonian cycle in G. Now I just have to close all my square brackets, there we go. And there beautifully you'll see here 1, 2, 3, 4, 5, 6, 7, 8, 9, back to 1. We see this Hamiltonian cycle. So create some of your own, very nice to play with all of these inside of Mathematica. And what you can also do, if you're here on the desktop version I'm going to hover there for instance and I'm going to click on that and there we have the documentation open up for us. You can read all about and play around with this one. This is the Stodecahedron here and you can find its Hamiltonian cycle or the first Hamiltonian cycle. And if you just looked at this Stodecahedron you might not think that there would be a closed Hamiltonian walk in there but lo and behold there it is. So let's increase the screen size a bit if we can, there we go. So click on that, remember there's such excellent documentation when it comes to Mathematica, have a look around.