Eulerian Graphs

An Eulerian graph is a graph in which it is possible to traverse every edge exactly once and return to the starting vertex. This concept is named after the mathematician Leonhard Euler, who solved the famous Seven Bridges of Königsberg problem in 1736.

Key Definitions

Euler Graph − A connected graph G is called an Euler graph if there is a closed trail (circuit) that includes every edge of the graph G exactly once.

Euler Path − An Euler path is a path that uses every edge of a graph exactly once. An Euler path starts and ends at different vertices.

Euler Circuit − An Euler circuit is a circuit that uses every edge of a graph exactly once. An Euler circuit always starts and ends at the same vertex.

Conditions for Eulerian Graphs

The following conditions determine whether a connected graph is Eulerian −

A connected graph G is Eulerian if and only if its edge set can be decomposed into cycles.

Example: Euler Graph

The above graph is an Euler graph. The trail a →1→ b →2→ c →3→ d →4→ e →5→ c →6→ f →7→ g covers all the edges of the graph. Every vertex has an even degree, satisfying the Eulerian condition.

Example: Non-Euler Graph

This graph is not an Euler graph because vertices a and d each have degree 3 (odd degree). However, since exactly two vertices have odd degree, an Euler path (starting at a and ending at d, or vice versa) does exist − but no Euler circuit is possible.

Conclusion

A connected graph has an Euler circuit if all vertices have even degree, and an Euler path if exactly two vertices have odd degree. If more than two vertices have odd degree, no Eulerian trail of any kind exists.