Euler Paths and Circuits Explained

Published: 24 February 2021
on channel: Study Force

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An Euler path is a path that travels through every edge of a graph once and only once.
Each edge must be traveled and no edge can be retraced.

An Euler circuit is a circuit that travels through every edge of a graph once and only once.
Like all circuits, an Euler circuit must begin and end at the same vertex.

Euler's Theorem (for connected graphs):
a. If a graph has exactly two odd vertices, then it has at least one Euler path, but no Euler circuit. Each Euler path must start at one of the odd vertices and end at the other one.
b. If a graph has no odd vertices (all even vertices), it has at least one Euler circuit (which, by definition, is also an Euler path). An Euler circuit can start and end at any vertex.
c. If a graph has more than two odd vertices, then it has no Euler paths and no Euler circuits.

Q. Given the graph in the figure:

a) Explain why it has at least one Euler path.

b) Use trial-and-error to find one such path

Q. Is it possible to walk across all 7 bridges without having to re-cross any of them?

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