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Can $C_n$ has $2*n$ Euler circuits and $3*n$ Hamilton paths in the Cyclic graphs?

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Bt $C_n$ you mean the cycle with $n$ vertices, right?

A cycle on $n$ vertices has $2n$ Euler circuits: Each circuit is uniquely determined by the starting vertex ($n$ choices) and then the direction (2 choices).

Each Hamiltonian path is also uniquely determined by the starting vertex $n$ choices and then the direction 2 choices. So there are $2n$ Hamiltonian paths.

Each Hamiltonian cycle is also uniquely determined by the starting vertex $n$ choices and then the direction 2 choices. So there are $2n$ Hamiltonian cycles.

So to answer your question in short, yes there are $2n$ Eulerian circuits, no there are only $2n$ Hamiltonian paths.

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  • $\begingroup$ Thanks @mike for the explanation :) $\endgroup$ – June Dec 18 '18 at 20:18

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