Eulerian Graph Maps

Euler Graphs. Consider the following road map . The explorer's Problem An explorer wants to explore all the routes between a number of cities. Can a tour be found which traverses each route only 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.

Notice that if a graph has an Eulerian path that is not a circuit it is generally not considered an Eulerian graph, although some authors will call it such. So in any reference you read, be sure to check that definition that is used! Using a recent road map, it appears that an Eulerian circuit exists in New York City, not including the

The Knigsberg Bridge Problem and Eulerian Graphs Figure 9.4.1 Map of Knigsberg A map of the Prussian city of Knigsberg circa 1735 in Figure 9.4.1 shows that there were seven bridges connecting the four land masses that made up the city. The legend of this problem states that the citizens of Knigsberg searched in vain for a walking

Lemma 1 If G is Eulerian, then every node in G has even degree. Proof Let G V, E be an Eulerian graph and let C be an Eulerian circuit in G.Fix any node v.If we trace through circuit C, we will enter v the same number of times that we leave it. This means that the number of edges incident to v that are a part of C is even. Since C contains every edge in the graph exactly once, this

An Eulerian trail, note 1 or Euler walk, in an undirected graph is a walk that uses each edge exactly once. If such a walk exists, the graph is called traversable or semi-eulerian. 3An Eulerian cycle, note 1 also called an Eulerian circuit or Euler tour, in an undirected graph is a cycle that uses each edge exactly once. If such a cycle exists, the graph is called Eulerian or unicursal. 4

An Eulerian graph is a graph containing an Eulerian cycle. The numbers of Eulerian graphs with n1, 2, nodes are 1, 1, 2, 3, 7, 15, 52, 236, OEIS A133736, the first few of which are illustrated above. The corresponding numbers of connected Eulerian graphs are 1, 0, 1, 1, 4, 8, 37, 184, 1782, OEIS A003049 Robinson 1969 Liskovec 1972 Harary and Palmer 1973, p. 117, the first

Eulerian Graphs Figure 9292PageIndex192 A map of Koenigsberg, circa 1735 Figure 9292PageIndex292 A multigraph for the bridges of Koenigsberg. Using a recent road map, it appears that an Eulerian circuit exists in New York City, not including the small islands that belong to the city. Lowell, Massachusetts, is located at the confluence

Eulerian Paths, Circuits, Graphs. An Eulerian path through a graph is a path whose edge list contains each edge of the graph exactly once. If the path is a circuit, then it is called an Eulerian circuit. An Eulerian graph is a graph that possesses a Eulerian circuit. Example 13.4.5. An Eulerian Graph.

A key example of planar graphs is a map where every country is a node and the edges represent having shared borders . Four Color Theorem An undirected graph has an Eulerian path if and only if exactly zero or two vertices have odd degree . Euler Path Example 2 1 3 4. History of the ProblemSeven Bridges of

An Eulerian graph is a graph that possesses an Eulerian circuit. Example 9.4.5 An Eulerian Graph. Without tracing any paths, we can be sure that the graph below has an Eulerian circuit because all vertices have an even degree. This follows from the following theorem. Figure 9.4.6 An Eulerian graph. Theorem 9.4.7 Euler's Theorem General Case.