Finding a Hamiltonian Path through the complete graph on 37 vertices: $K_{37}$ I'm planning on making a fiber art $K_{37}$ (like the one I laser etched with help: K37: The complete graph on 37 nodes, svg). To accomplish this, the plan is to construct 37 pegs equally spaced in a annulus made from wood and then to string yarn between them. Since $K_{37}$ is a complete graph, it has a Hamiltonian path. My knowledge of computational graph theory is nil, and I am aware that this is not exactly an easy problem (see wikipedia: Hamiltonian Path Problem, for instance)
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.