how to draw a graph with 5 vertices which is:
a)Eulerian cycle but not Hamiltonnian cycle. b)Hamiltonnian cycle but not Eulerian cycle. c)Eulerian cycle and Hamiltonnian cycle.
how to draw a graph with 5 vertices which is:
a)Eulerian cycle but not Hamiltonnian cycle. b)Hamiltonnian cycle but not Eulerian cycle. c)Eulerian cycle and Hamiltonnian cycle.
Two days later, this is probably worth an answer.
If your points are A, B, C, D, E then the edges AB, BC, CD, DE, EA clearly provide one of the smallest graphs which has both a Eulerian cycle and Hamiltonian cycle.
Add another edge, for example AC, and you have a graph which has the same Hamiltonian cycle but not an Eulerian cycle, since A and C have an odd number of edges.
Alternatively, the edges AB, BC, CA, AD, DE, EA, provide a Eulerian cycle but not a Hamiltonian cycle, since you can only get from B to D via A, and you cannot get back without visiting A again.
Incidentally, the neat Flash game at http://neamar.fr/Res/Icosien/ asks you to find 20 Eulerian cycle and Hamiltonian cycles for given graphs.