I found myself having to find a Hamilton path in a graph of 200 vertices (what you called "nodes"). My first idea was to use the Hamilton path solver in Maxima ( http://maxima.sourceforge.net/docs/manual/de/maxima_50.html#IDX2257 ), but it wasn't able to solve it. Luckily, a few years ago, I used to use SAT solvers to solve difficult logic grid puzzles (because I had a friend who kept on sending them to me!), so wrote a program to write a SAT model for the graph. To my delight, when I fed the model to an open source SAT solver (MiniSat), it was able to resolve it in less than 10 minutes (I didn't time it I'm afraid - I started it, and it had finished when I came back).
Disclaimer: I am not a "theoretical computer scientist", nor even somebody who has ever done a Hamilton cycle before - I landed here after a Google search and thought I'd share what I did over the weekend. However, for a beginner, beating the mathematicians who wrote the Graphs package in Maxima isn't bad - and suggests that if you need to find a Hamilton path (or cycle) in a large graph, building a SAT model is an approach worth considering.