Euler's path same as Hamiltonian cycle

[v-v]

Homework Statement

Find (classify) all graphs with n vertices who's Euler's path is the same as their Hamiltonian cycle.

The Attempt at a Solution

I'd say that any regular graphs with a degree greater than n/2 (Dirac's Theorem) with and n divisible by 2 has both and Euler's path and a Hamiltonian cycle. But that doesn't mean they're the same path on that graph. Also that probably does not define all graphs where that's the case.

 

[grief]

Think about what each of the definitions mean. It seems that it's a pretty strong condition that a path touching each vertex once also covers each edge once. Specifically, it means the graph needs to have only the edges in the hamiltonian cycle, and no others. See if you can go from here to find a characterization of such graphs.

 

[v-v]

I don't think there are other graphs but the Cycle Graph that meets that criteria :\

 

[JohnnyDR]

Try using an app for iPhone called GraphSolver, you can create your graphs and solve eulerian circuit, paths, hamilton and stuff.

Here is the link maybe it can help you:

http://itunes.apple.com/us/app/graphsolver/id425914392?mt=8&ls=1