Question

Give an example of a graph which is Eulerian but not Hamiltonian.​

Answer 1

Answer:

The full bipartite graph $K_{2,4}$ is non-Hamiltonian but has an Eulerian circuit.

Step-by-step explanation:

An Euler circuit starts and ends at the same vertex and uses each vertex exactly once (Eulerian) (circuit). It's worth noting that an Eulerian circuit can visit vertices multiple times.

Each vertex is visited exactly once by a Hamiltonian cycle.

Here's a graph with an Eulerian circuit (start at any vertex and "draw" a figure-eight), but no Hamiltonian cycle since any path that hits every vertex would have to visit the middle vertex too frequently:

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