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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.

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please help me answer this question –  user6788 Feb 8 '11 at 12:10
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Write down the definitions of a Eulerian and Hamiltonian cycle and it should be quite easy with some trials. Do you remember the characterisation of Eulerian graphs ? –  Klaus Feb 8 '11 at 12:17
    
Eulerian cycle-accessible all node once and again,compulsory cross every node while Hamiltonian cycle-node must be pass through once only ,can skip node. –  user6788 Feb 9 '11 at 11:10
    
No, Eulerian cycles use all edges and return to start. Hamiltonian cycles use all vertices once each and return to start. –  Ross Millikan Feb 10 '11 at 19:17
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1 Answer 1

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.

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