Definition:Semi-Eulerian Graph

Definition

A graph is called semi-Eulerian if and only if it contains an Eulerian trail.

Note that the definition of graph here includes:

• Simple graph
• Loop-graph
• Multigraph
• Loop-multigraph.

Note that an Eulerian graph is also semi-Eulerian, as an Eulerian circuit is still a path, and therefore an Eulerian trail.

Examples

Arbitrary Example

The following is a semi-Eulerian graph:

An example of an Eulerian trail is:

$A \to B \to C \to D \to E \to B \to C \to A \to E$

Also known as

A semi-Eulerian graph is also called a traversable graph.

Some sources have transversable graph but it is suspected that this is a mistake.

Also see

• Definition:Eulerian Graph

• Eulerian Graph is Semi-Eulerian

• Results about semi-Eulerian graphs can be found here.

Source of Name

This entry was named for Leonhard Paul Euler.

Sources

• 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): $\S 3.1$: The Königsberg Bridge Problem: An Introduction to Eulerian Graphs
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Eulerian graph
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Eulerian graph
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): transversable graph
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): traversable graph