Definition:Eulerian Graph

Definition

A loop-multigraph or loop-multidigraph is called Eulerian if and only if it contains an Eulerian circuit.

Examples

Arbitrary Example

The following is an Eulerian graph:

An example of an Eulerian circuit is:

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

Also known as

An Eulerian graph is also called a unicursal graph.

Also see

• Eulerian Graph is Semi-Eulerian

• Characteristics of Eulerian Graph
• Definition:Semi-Eulerian Graph

• Definition:Hamiltonian Graph

• Results about Eulerian graphs can be found here.

Source of Name

This entry was named for Leonhard Paul Euler.

Historical Note

The term Eulerian graph arises from Leonhard Paul Euler's work on the Bridges of Königsberg problem.

Linguistic Note

After the pronunciation of Euler as Oiler, the word Eulerian is pronounced something like oy-leer-ee-an.

The term unicursal, derives from the prefix uni-, an adjectival form of one and cursal, which derives from the Latin cursor, meaning runner.

Hence a unicursal graph can be understood as a graph which can be drawn in a single run.

Sources

• 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): $\S 3.1$: The Königsberg Bridge Problem: An Introduction to Eulerian Graphs: Theorem $3.1$
• 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): Eulerian graph