Let be $G=(V,E)$, where $V=\{1,\ldots,n\}$ and $E=\{\{i,j\}\subset V;ij\leq k\}$ and $k<n$.
For which values of $k\geq 2$, can we count explicitly the number of Hamiltonian paths in $G$ ?
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S. Kitaev defines Path schemes $P(n,M)$ as graphs with vertex set $\{1,2,\dots,n\}$ and edges $(i,j)$ iff $ij\in M$. Hamiltonian graphs on path schemes were mentioned in "On uniquely kdetermined permutations" by S. Avgustinovich and S. Kitaev. The formula is not simple even in the case where $M=\{1,2\}$ (here), but I guess it depends on what kind of formula you are looking for. 


Explicit values for $k\leq 9$ and small $n$ are given in the OEIS: k=2: http://oeis.org/A003274 (contains some references and a generating function) 

