Hello, and welcome to the next lesson of the graphs class devoted to Eulerian and Hamiltonian cycles. It's usual we start by providing some motivational puzzle, which in this case is actually a simplified version of an important practical genome assembly problem. In this case, this is probably even an oversimplified version. So, in the Genome Assembly Problem, where we can to reconstruct a long genome. A genome can be considered in this case as a long string consisting of symbols A, C, G and T. It is extremely long. Usually it consists of millions or even billions symbols. The problem is that we can not read the genome. What we can do is actually what current sequencing technologies allow us to do, is to read a short fragment of a genome. And we can do, we can read many such short substrings of a genome. And they are called threes. They are usually roughly several hundreds symbols long. Then our goal the problem of genome assembly is to reconstruct the initial stream from many of its short pieces. So in this case as a toy example of this problem, we can see there's a following problem. Our goal is to find a string whose all three sub strings are the view in one. So in this case we are given 8 strings of lengths 3, and we are looking for a string who is all 3 strings, are exactly the ones that are shown here. At this point it should not be clear how it is all related to Hamiltonian cycles or Eulerian cycles. But it will be clear in several minutes. Okay. So let me first introduce Eulerian cycles. As you probably already know an Eulerian cycle in a graph visits each edge exactly once, okay? Let me make several remarks about this definition. First of all it works for directed and undirected graphs. The only difference is that in directed graphs by a cycle or a path would mean a path or a cycle that follows the directions of all the edges, okay. So in undirected case we can traverse any edge in both directions, in directed case we can only reverse a match in its direction, okay? So the results are related to notion of a neighboring path, and the difference between a path and the cycle is that a cycle should start and end in exactly the same node while a path might start at some node and stop at some other node, okay. This is an example of a Non-Eulerian graph, so in this case it is not possible to traverse all the edges. Either by a site or by a path. Okay? And this is actually a graph from the famous Konigsberg bridges puzzle where the graph theory actually started. Okay and this is an example of an Eulerian path, but we convince you that it is indeed an Eulerian well all the numbers here show actually in order of traversing all the edges in the graph. Let me just try to follow this path so we go here then we go there, then we go there, there, there, there, there, there, and there. Okay and we actually even stop at the same vertex where we have started so this graph contains en Eulerian cycle so as a natural question is of course what graphs do you have in Eulerian cycle and what graph do not have Eulerian cycles?