I understand the conditions necessary for a graph to have Eulerian and Hamiltonian paths. I could find examples for graphs that are Eulerian but not Hamiltonian. Can someone give me graphs that are non-Eulerian but are Hamiltonian?
Make a cycle on $4$ or more vertices. Then join two unjoined vertices with an edge. Then join two different unjoined vertices with an edge.
A $K_4$ is Halmiltonian but not Eulerian. You can't traverse every edge in this graph in a walk without repeating edges.