Euler Circuit and Paths Review

Front

What is an Euler Circuit?

Back

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

Front

What is an Euler Path?

Back

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

Front

What is the valence of a vertex?

Back

The valence (or degree) of a vertex is the number of edges connected to it.

Front

What are the conditions for a graph to have an Euler Circuit?

Back

A graph has an Euler Circuit if all vertices have even valence.

Front

What are the conditions for a graph to have an Euler Path?

Back

A graph has an Euler Path if it has exactly zero or two vertices of odd valence.

Front

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

Back

No, a graph cannot have an Euler Path if it has more than two vertices of odd valence.

Front

Can a graph with all vertices of even valence have an Euler Path?

Back

Yes, a graph with all vertices of even valence will also have an Euler Circuit, which implies it has an Euler Path.

Front

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

Back

Euler's Theorem provides the conditions under which a graph can have Euler Circuits and Paths, guiding the study of graph traversal.

Front

How can you determine if a graph has an Euler Circuit or Path by examining its vertices?

Back

Count the valence of each vertex: if all are even, it has an Euler Circuit; if exactly two are odd, it has an Euler Path.

Front

What is an example of a graph that has an Euler Circuit?

Back

A simple square (4 vertices, 4 edges) has an Euler Circuit as all vertices have even valence.