Euler Paths and Circuits

Front

What is a Euler Circuit?

Back

A Euler Circuit is a path in a graph that visits every edge exactly once and returns to the starting vertex.

Front

What is a Euler Path?

Back

A Euler Path is a path in a graph that visits every edge exactly once but does not necessarily return to the starting vertex.

Front

How can you determine if a graph has a Euler Circuit?

Back

A graph has a Euler Circuit if all vertices have even degrees.

Front

How can you determine if a graph has a Euler Path?

Back

A graph has a Euler Path if it has exactly 0 or 2 vertices of odd degree.

Front

What does it mean for a vertex to have an even degree?

Back

A vertex has an even degree if it is connected to an even number of edges.

Front

What does it mean for a vertex to have an odd degree?

Back

A vertex has an odd degree if it is connected to an odd number of edges.

Front

Can a graph have both a Euler Circuit and a Euler Path?

Back

Yes, if a graph has a Euler Circuit, it also has a Euler Path.

Front

What is the relationship between Euler Paths and Euler Circuits?

Back

Some Euler Paths are Euler Circuits, but not all Euler Paths are Euler Circuits.

Front

If a graph has 3 vertices of odd degree, can it have a Euler Path?

Back

No, a graph cannot have a Euler Path if it has more than 2 vertices of odd degree.

Front

What is the significance of Euler's Theorem in graph theory?

Back

Euler's Theorem provides conditions under which a graph can have Euler Paths and Circuits.