Does a graph that has an Eulerian circuit necessarily have a

Does a graph that has an Eulerian circuit necessarily have a Hamiltonian circuit? Explain your answer.

Solution

having a Euler circuit dos not always mean that it will have a hamiltonian circuit.

the graph has Euler circuit and Hamiltonian circuit.(each vertex is travelled once )

but the following graph has NO Euler circuit but it has a Hamiltonian Circuit

Degree of vertex is odd , Hence no Euler Circuit. while take a path , the vertex is travelled only once. Hence a Hamiltonian CIrcuit

.HEnce the graph having Euler circuit does not necessarily mean having a HAmiltonian graph