Find hamilton cycle in a directed graph reduced to sat problem

Question

I need to find a Hamiltonian cycle in a directed graph using propositional logic, and to solve it by sat solver. So after I couldn't find a working solution, I found a paper that describes how to construct a CNF formula to find an Hamiltonian path:

Xi,j - node j is in position i in the path

List of constraints:

1. Each node j must appear in the path x1j ∨ x2j ∨ · · · ∨ xnj - for every node j
2. No node j appears twice in the path ¬xij ∨ ¬xkj for all i, j, k with i 6= k.
3. Every position i on the path must be occupied - xi1 ∨ xi2 ∨ · · · ∨ xin for each i
4. No two nodes j and k occupy the same position in the path - ¬xij ∨ ¬xik for all i, j, k with j != k
5. Nonadjacent nodes i and j cannot be adjacent in the path - ¬xki ∨ ¬xk+1,j for all (i, j) !∈ E and k = 1, 2, . . . , n − 1.

My question is, how can I find Hamiltonian cycle using these constraints? I understand that I need to check if there's a cycle (v1==vn), that's one thing (But I got constraint 2). Second, it's a directed graph and I don't know how can I assure that the vertices would be in the right order of the edges, I thought about this:

Every two nodes must have edges - Xki ^ Xk+1j for each (i,j)∈ E and k = 1, 2, . . . , n − 1.

But it doesn't seem to work, any help would be appreciated.

EDIT

What I did was to add another constraint -

6. No edges from the set: (i,j)!∈ E that are the last and the first

Answer 1

Just add the constraint that $x_{1j}=x_{nj}$ for all $j$, and make a special exception to constraint 2 so that you omit constraint 2 in the case where $i=1$ and $k=n$.

Everything works fine for a directed graph. You just need to interpret constraint 5 appropriately.

Answer 2

I suppose practical SAT solvers might search for and remove them, but I'd still suggest not giving it redundant variables - if you do know v1==vn, then just replace each instance of vn with v1.
If n is the number of vertices then you'd actually have v0 == vn, rather than v1==vn.

You can remove any one of {1.,2.,3.,4.}, since the combination {3.,4.} or {1.,2.}
will force X or its transpose respectively to be the graph of a function, and
for functions from a finite set to itself, surjectivity is equivalent to injectivity.

Answer 3

I think all you need is to extend the fifth constraint to all k = 1, ...,n and where there is k+1 in the constraint, add (mod n) operator.

This ensures that there is an edge from the last vertex to the first in the path, meaning it's a cycle.