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Euler Paths and Circuits (5)
Euler paths trace every edge exactly once, regardless of how many times they visit each vertex. They
may or may not start and end at the same vertex. If they do start and end at the same vertex, they are
considered Euler circuits. Therefore, an Euler circuit is a “special kind” of Euler path.
Which of the vertex-edge graphs have Euler paths?
Which have Euler circuits?
What is the degree of each vertex?
Note: If the degrees of all the vertices on an Euler graph are even, then the graph has an Euler circuit. If
the degrees of exactly two of the vertices on an Euler graph are odd, then the graph has an Euler path.
Solve Problems using Euler Paths and Circuits (5)
Peter’s Cat
Peter’s cat was lost. Peter canvassed his neighborhood with a flyer describing his missing cat. It was
important that Peter visit every street in his neighborhood as soon as possible. On the vertex-edge graph
below, trace a route he might take.
Is he able to start at one vertex, travel every edge only once, and return to his starting vertex? __________
If yes, he has made an Euler circuit. Record three possible routes below.
© Rodel Foundation of Arizona, 2009