[Math] Why is repeated vertices allowed in Euler path

discrete mathematicsgraph theory

As wikipedia defines path as:

A path is a trail in which all vertices (except possibly the first and
last) are distinct.

if we cannot repeat a vertex in a path then how can we repeat it in Euler path ? ( which is a path in which every edge is visited exactly once )

For example:

Example Graph

In the above graph A-B-C-D-E-F-C-A-F is an Euler path but as you can see A, F and C vertices are visited twice which is not allowed in a path then how can you say its a path ?

Thanks for your precious time.

Best Answer

An 'eulerian path' need not be a 'path'. As already mentioned by someone, the exact term should be eulerian trail. The example given in the question itself clarifies this fact. The trail given in the example is an 'eulerian path', but not a path. But it is a trail certainly. So, if a trail is an eulerian path, that does not mean that it should be a path at the first place.

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