# Eulerian path

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## Eulerian path

[ȯi′ler·ē·ən ′path]
(mathematics)
A path that traverses each of the lines in a graph exactly once.
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There is an Euler path, for example, that sweeps alternately from left to right and from right to left through the edges belonging to each successive generation, as shown in Figure 6.
For the inductive step, observe that if there is an Euler path for a graph with n generations and the nth row of edges sweeps from left to right (or right to left) then constructing another row of edges (and vertices) sweeping the opposite way will produce the graph with n + 1 generations.
A well-known theorem states that a graph has an Euler path if and only if the number of nodes with odd degree is two or fewer.
Anil comments "Eric Iverson arbitrarily assumed that R is not an Euler Path letter.
A well-known theorem in graph theory states that a network contains an Euler Path (a path that traverses the network, once only along each link) if and only if it has at most two nodes with an odd number of links.
If one posits a sans-serif alphabet, the Euler Path letters (ones that can be traced out without lifting pencil from paper) are BCDGIJLMNOPQSUVWZ.
One can also construct words with no Euler Path letters, the longest being THEREAFTER.

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