# Euler Circuit

Bollobas (1979) claimed that one of the oldest problems regarding graphs is the Konigsburg bridge problem. The problem reduces to: • Is there a closed path in this graph that uses each edge exactly once? • Such a path is called an Euler circuit of the graph. • A graph that has an Euler circuit must have all vertices of even degree. • A finite connected graph in which every vertex has even degree has an Euler circuit. • A simple path that contains all the edges in a graph G is called an Euler path.

According to Math Forum, an Euler circuit is a graph traversal beginning and ending at the same vertex and utilizing each edge precisely once (http://www. mathforum. org). Coming across an Euler circuit in a directed or an undirected graph is a somewhat easy task. However, what about graphs where a number of the edges are directed and several undirected? Math Forum said that an undirected edge could merely be traveled in one direction. Nevertheless, from time to time any preference of direction for the undirected edges might not generate an Euler circuit.

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Bollobas (1979) asserted that an Eulerian trail that begins and ends at the identical graph vertex. Hence, as stated earlier, it is a graph cycle that utilizes every graph edge precisely once. The expression Eulerian cycle is likewise employed synonymously with Eulerian circuit. For technical explanations, Eulerian circuits are mathematically easier to learn compared to the Hamiltonian circuits (Bollobas, 1979). Bollobas (1979) further said that as a simplification of the Konigsberg bridge problem, Euler demonstrated, without evidence, that a connected graph has an Eulerian circuit if it has no graph vertices of odd degree.

How to Find an Euler Circuit As asserted by Bollobas (1979), if a graph is connected, and if every vertex has even degree, then there is an Euler circuit in the graph. Buried in that evidence is an explanation of an algorithm for finding such a circuit as follows: (a) First, pick a vertex to the “start vertex. ” (b) Find at random a cycle that begins and ends at the start vertex. Mark all edges on this cycle. This is now your current circuit. (c) If there is not a vertex on the current circuit that is incident to an unmarked edge, you are done.

If there is such a vertex, find a random cycle using unmarked edges that begin and ends at this vertex. Mark the edges in this cycle as you find it. Splice this cycle into the current circuit to make a new, larger current circuit that begins and ends at the start vertex. Repeat this step. Illustration of Euler Circuit Planet Math (http://planetmath. org) stated that an Euler circuit is a connected graph such that beginning at a vertex , one can traverse along each edge of the graph once to each of the other vertices and return to vertex .

Therefore, an Euler circuit is an Euler path that is a circuit. Consequently, employing the properties of odd and even degree vertices given in the definition of an Euler path, an Euler circuit exists if and only if each vertex of the graph has an even degree. This graph is an Euler circuit as all vertices have degree 2. Euler’s Theorems Euler has three theorems as follows (http://instruction. elgin. edu): Euler’s Theorem 1: If a graph has any vertices of odd degree, then it can’t have any Euler circuit.

If a graph is connected and every vertex has an even degree, then it has at least one Euler circuit (usually more). Euler’s Theorem 2: If a graph has more than two vertices of odd degree, then it cannot have an Euler path. If a graph is connected and has exactly two vertices of odd degree, then is has at least one Euler path. Any such path must start at one of the odd degree vertices and must end at the other odd degree vertex. Euler’s Theorem 3: The total number of degrees of all vertices of a graph is an even number (twice the number of edges).

In every graph, the number of vertices of odd degree must be even. Fleury’s Algorithm in Relation to Euler Circuit Bollobas (1979) claimed that Fleury’s algorithm is a well-designed, yet ineffective, technique of producing Eulerian circuit. The single Platonic solid having an Eulerian circuit is the octahedron that has Schlafli symbol; all other Platonic graphs have odd degree sequences (Bollobas, 1979). Likewise, the lone Eulerian Archimedean solids are the small rhombicosidodecahedron, icosidodecahedron, cuboctahedron, and small rhombicuboctahedron.

Fleury’s algorithm provides us the way for discovering effective means in which to go over every the edge of a graph, on condition that the graph has either zero or two vertices of odd degree (http://instruction. elgin. edu). If the graph has over two vertices of odd degree, we will have to retrace a number of the edges so as to navigate all edges at least once. With the aim of minimizing the retracing, we should change the vertices of odd degree to even degree by means of adding additional edges. The process of eliminating odd vertices by adding additional edges is dubbed as “eulerizing” the graph. This means that the edges that are added in this process should be duplicates of existing edges. #

References

Bollobas, B. (1979). Graph Theory: An Introductory Course. New York: Springer-Verlag. “Euler Circuits. ” Retrieved October 23, 2006 at http://instruction. elgin. edu. Math Forum. Problem D: Euler Circuits. Retrieved October 23, 2006 at http://www. mathforum. org Planetmath. org. Euler Circuit. Retrieved October 23, 2006 at http://planetmath. org

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