Euler circle and Hamilton cycles Given $K_n$ define sequence $S_4$: writing all vertices in order of traversing it by Euler circle How many distinct Hamilton cycles are in this sequence $S$?

It can certainly vary. As an Eulerean path uses all $\frac{n(n-1)}{2}$ edges and a Hamiltonian path uses $n$, you can't have more than $\frac{n-1}{2}$. Can you have that many? There aren't any Eulerean paths if $n$ is even and $>2$. Depending upon your definition, I can define an Eulearean path that has no segment that is Hamiltonian