Eulerian and hamiltonian graphs pdf

Being a circuit, it must start and end at the same vertex. Eulerian path and circuit an eulerian path also called an euler path and an eulerian trail in a graph is a path which uses every edge exactly once. Note that if g contains exactly two odd vertices, then the fleurys algorithm. A graph is hamiltonian if it has a cycle that contains all the vertices of a graph. Algebraic graph, hamiltonian and eulerian algebraic graphs. Hamiltonian paths and cycles 2 remark in contrast to the situation with euler circuits and euler trails, there does not appear to be an efficient algorithm to determine whether a graph has a hamiltonian. The following problem, often referred to as the bridges of konigsberg problem, was first solved by euler in the eighteenth century. Eulerian cycles of a graph g translate into hamiltonian cycles of lg. Euler and hamiltonian paths and circuits lumen learning. Chapter 1 directed graphs, paths recall that a directed graph, g, is a pair g v. Graph theory eulerian and hamiltonian graphs ulsites.

One of the most notable instances is their connection with the fourcolor conjecture. A connected graph g is eulerian if and only if the degree of each of its vertices is even. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Any hamiltonian cycle can be converted to a hamiltonian path by removing one of its edges, but a hamiltonian path can be extended to hamiltonian cycle only if its endpoints are adjacent. Eulerian and hamiltonian graph mathematics stack exchange. Eulerian and hamiltoniangraphs there are many games and puzzles which can be analysed by graph theoretic concepts. Eulerian and hamiltonian graphs stanford university. If there are vertices of odd degree, halt and return. The konisberg bridge problem konisberg was a town in prussia, divided in four land regions by the river pregel. The seven bridges of konigsberg problem is also considered. In fact, the two early discoveries which led to the existence of graphs arose from. Eulerian and hamiltonian cycles eulerian paths and cycles question.

Learning outcomes at the end of this section you will. In fact, the solution by leonhard euler switzerland, 170783 of the koenigsberg bridge problem is considered by many to represent the birth of graph theory. A hamiltonian path also visits every vertex once with no repeats, but does. A famous problem which goes back to euler asks for what graphs g is there a closed walk which uses every edge exactly once. Eulerian and hamiltonian graphs the subject of graph traversals has a long history. Finding an euler path there are several ways to find an euler path in a given graph. Throughout this text, we will encounter a number of them. Chapter 1 directed graphs, paths recall that a directed graph, g, is a pair g v,e, where e. You can verify this yourself by trying to find an eulerian trail in both graphs. This condition for a graph to be hamiltonian is shown to imply the wellknown. One of the most notable instances is their connection with the fourcolor.

In this chapter, we present several structure theorems for these graphs. Some applications of eulerian graphs 3 thus a graph is a discrete structure that gives a representation of a finite set of objects and certain relation among some or all objects in the set. An euler trail in a graph g is a spanning trail in g that. Line graphs may have other hamiltonian cycles that do not. They were first discussed by leonhard euler while solving the famous seven. Nov 03, 2015 a brief explanation of euler and hamiltonian paths and circuits. Eulerian graphs and related topics, volume 1 1st edition. A connected graph is called eulerian if it has a closed trail containing all edges of the graph. The konigsberg bridge problem and eulerian graphs figure 9. Similarly, an eulerian circuit or eulerian cycle is an eulerian trail that starts and ends on the same vertex. All hamiltonian graphs are biconnected, but a biconnected graph need not be hamiltonian see, for example, the petersen graph. Graph theory is an area of mathematics that has found many applications in a variety of disciplines. Eulerian and hamiltonian graphs aim to introduce eulerian and hamiltonian graphs. Circuit decompositions of eulerian graphs genghua fan1 department of mathematics, arizona state university, tempe, arizona 85287 and cunquan zhang2 department of mathematics, west virginia university, morgantown, west virginia 265066310 received february 25, 1991 let g be an eulerian graph.

We shall now express the notion of a graph and certain terms related to graphs in a little more rigorous way. A hamiltonian circuit is a circuit that visits every vertex once with no repeats. An euler circuit or eulerian circuit in a graph \g\ is a simple circuit that contains every edge of \g\ reminder. The subject of graph traversals has a long history. An eulerian graph is a connected graph that has an eulerian circuit. Hamiltonian and eulerian graphs eulerian graphs if g has a trail v 1, v 2, v k so that each edge of g is represented exactly once in the trail, then we call the resulting trail an eulerian trail. If an edge has a vertex of degree d 1 at one end and a vertex of degree d 2 at the other, what is the degree of its corresponding vertex in the line graph. An eulerian circuit also called an eulerian cycle in a graph is an eulerian path that starts and. The graph on the left is not eulerian as there are two vertices with odd degree, while the graph on the right is eulerian since each vertex has an even degree. An undirected graph has an euler circuit iff it is connected and has zero vertices of odd degree. The duality between atrails in plane eulerian graphs and hamiltonian cycles in plane cubic graphs.

This tutorial offers a brief introduction to the fundamentals of graph theory. An eulerian circuit also called an eulerian cycle in a. It was euler who showed in 1736 that the celebrated konigsberg bridge problem has no solution. A connected graph g is hamiltonian if there is a cycle which includes every vertex of g. A brief explanation of euler and hamiltonian paths and circuits. The study of hamiltonian graphs has been important throughout the history of graph theory. Graph theory eulerian and hamiltonian graphs aim to introduce eulerian and hamiltonian graphs. In fact, the solution by leonhard euler switzerland, 170783 of the koenigsberg bridge. If the trail is really a circuit, then we say it is an eulerian circuit. Circuit decompositions of eulerian graphs genghua fan1 department of mathematics, arizona state university, tempe, arizona 85287 and cunquan zhang2 department of mathematics, west virginia. A trail contains all edges of g is called an euler trail and a closed euler trial is called an euler. If an edge has a vertex of degree d 1 at one end and a vertex of degree d 2 at the other, what is the degree of its corresponding. In graph theory, an eulerian trail or eulerian path is a trail in a finite graph that visits every edge exactly once allowing for revisiting vertices. You will only be able to find an eulerian trail in the graph on the right.

In fact, the two early discoveries which led to the existence of graphs arose from puzzles, namely, the konigsberg bridge problem and hamiltonian game, and these puzzles. The problem is to find a tour through the town that crosses each bridge exactly once. Eulerian graphs the following problem, often referred to as the bridges of k. This eulerian cycle corresponds to a hamiltonian cycle in the line graph lg, so the line graph of every eulerian graph is hamiltonian graph. If a graph has such a circuit, we say it is eulerian. In fact, the solution by leonhard euler switzerland, 170783 of the konigsberg bridge problem is considered by many to represent the birth of graph theory. But there are certain criteria which rule out the existence of a hamiltonian circuit in a graph, such as if there is a vertex of degree one in a graph then it is impossible for it to have.

Math 3012 lecture 11 eulerian and hamiltonian graphs. Eulerian and hamiltonian cycles jeangallier april16,20. Unlike euler paths and circuits, there is no simple necessary and sufficient criteria to determine if there are any hamiltonian paths or circuits in a graph. This assumes the viewer has some basic background in graph theory. The study of eulerian graphs was initiated in the 18th century, and that of hamiltonian graphs in the 19th century. Fleurys algorithm produces an euler tour in an eulerian graph. Mathematics euler and hamiltonian paths geeksforgeeks. An euler cycle or circuit is a cycle that traverses every edge of a graph exactly once.

A connected graph g is eulerian if there is a closed trail which includes every edge of g, such a trail is called an eulerian trail. A graph g is said to be hamiltonian graph if g has a hamiltonian circuit. These graphs possess rich structure, and hence their study is a very fertile field of research. Hamiltonian circuit is a circuit which passes through each vertex exactly once except initial vertex. Eulerian graphs and semieulerian graphs mathonline. Written in a readerfriendly style, it covers the types of graphs, their properties, trees, graph traversability, and the concepts of. The regions were connected with seven bridges as shown in figure 1a. Math 3012 lecture 11 eulerian and hamiltonian graphs lus pereira georgia tech september 14, 2018. An obvious and simple necessary condition is that any hamiltonian digraph must be strongly connected. Know what an eulerian graph is, know what a hamiltonian graph is. For the moment, take my word on that but as the course progresses, this will make more and more sense to you. A graph is said to be eulerian if it contains an eulerian circuit. These graphs possess rich structure, and hence their study is a very fertile field of research for graph theorists. Know what an eulerian graph is, know what a hamiltonian graph.