Finding a Hamiltonian Path through the complete graph on 37 vertices: $K_{37}$
It seems that you're trying to construct the graph $K_{37}$ from string and nails without cutting the string.  For this, you don't want a Hamiltonian path (a path that visits every vertex exactly once) but an Euler trail (a walk that visits every edge exactly once).

In the case of a complete graph, finding an Euler trail is trivial: start at any vertex and move from vertex to vertex without repeating an edge. As long as the last thing you do is returning to the initial vertex, you've created an Euler trail.  (This is essentially Fleury's algorithm except that checking you've not disconnected the graph is trivial for a complete graph.