Euler Circuits
Euler Circuits
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Euler Circuits
Leonhard Euler was perhaps one of the most prolific mathematicians of the last several centuries. The method he devised to travel a route in an efficient manner remains a highly used in the public sector. A Euler circuit is one that covers each edge of a graph once, but not more. Euler defined rules in two theorems; Theorem 1-If a graph has any vertices of odd degree, then it cannot have a Euler Circuit and If a graph has all even vertices, then it has at least one Euler Circuit - usually more. Theorem 2-If a graph has more than 2 vertices of odd degree, then if cannot have a Euler Path, and, if a graph is connected and has exactly 2 vertices of odd degree, then it has at least one Euler path. This path must start at one of the odd-degree vertices and end at the other.
Now that Euler circuits have been defined, it can be discussed how to find them. When considering a graph, its valences must be considered. The valences are the number of edges meeting at the vertex, or the point in a graph at which one or more points terminate. Keeping this in mind, it is possible to determine if a circuit is Euler in nature by seeing if the valences are even. If this is true, then it is a Euler circuit. If you wish to Eulerize a graph, it is simple to do. Add an edge that is a duplicate of an existing edge, thus making the valences even.