# Hamilton path and Euler circuits in Cycle graph

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

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.