Name __________________________________
An Euler path travels along every edge in the graph exactly once. Vertices may be revisited, but
edges may not be repeated. An Euler circuit is an Euler path that ends where it begins.
Peter’s cat was lost. He 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.
1.
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2.
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3.
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© Rodel Foundation of Arizona, 2009

Name __________________________________
An Euler path travels along every edge in the graph exactly once. Vertices may be revisited, but
edges may not be repeated. An Euler circuit is an Euler path that ends where it begins.
© Rodel Foundation of Arizona, 2009

Name __________________________________
A Hamilton path is a path that starts at a vertex in the graph and visits every other vertex of the graph exactly once. Edges along this path may be repeated. A Hamilton circuit is a Hamilton path that ends at the starting vertex. The shortest route may or may not be a Hamilton path.
Depending upon the constraints of a problem, each vertex may not need to be visited.
\If the park ranger is required to visit every location on the vertex-edge graph below, what route should he take? Where should he begin and end his trip?
1.
One possible Hamilton path is: Prospector-Tent-Coyotes-Snakes-Javelinas-Watering Hole-
Cacti-Cave Creek Canyon. Can you find other Hamilton paths?
2.
Is it possible to start at one vertex (site) on the vertex-edge graph and visit every other vertex just once and return to the starting vertex? If it is possible, name that circuit.
3.
What is the shortest route between Cave Creek Canyon and the Tent?
© Rodel Foundation of Arizona, 2009

Name __________________________________
A Hamilton path is a path that starts at a vertex in the graph and visits every other vertex of the graph exactly once. Edges along this path may be repeated. A Hamilton circuit is a Hamilton path that ends at the starting vertex. The shortest route may or may not be a Hamilton path.
Depending upon the constraints of a problem, each vertex may not need to be visited.
© Rodel Foundation of Arizona, 2009