Question

A graph is said to be Eulerian if it contains a closed trail that includes every:

• A. Vertex exactly once.
• B. Edge exactly once.
• C. Vertex at least once.
• D. Edge at most once.

Hint:

A connected graph is Eulerian if and only if every vertex in the graph has an even degree. This is a very fast way to check if a graph contains an Eulerian circuit.

Answer 1

The Correct Option is B

Concept: The concept of Eulerian graphs originated from the "Seven Bridges of Königsberg" problem solved by Leonhard Euler.
• Eulerian Trail: A path that visits every edge in the graph exactly once.
• Eulerian Circuit: A trail that is "closed" (starts and ends at the same vertex) and visits every edge exactly once.
• Eulerian Graph: A graph that possesses an Eulerian circuit.

Step 1: Understanding the requirement for edges.
By definition, an Eulerian path or circuit is concerned with traversing every edge of the graph. The critical constraint is that each edge must be used exactly one time.

Step 2: Contrasting with Hamiltonian paths.
It is a common point of confusion to mix Eulerian and Hamiltonian definitions:
• Eulerian: Every edge exactly once.
• Hamiltonian: Every vertex exactly once. Since the question specifically asks for the definition of an Eulerian graph, we focus on the edges.

Step 3: Confirming the "closed" nature.
The question mentions a "closed trail." A closed trail is a circuit. For this circuit to make the graph Eulerian, it must cover every edge exactly once.