In a weighted undirected graph, how do I find a fixed length path between two nodes?

You can always go with a Breadth First Search: http://en.m.wikipedia.org/wiki/Breadth-first%5Fsearch Where it's normally used to find the shortest path, you should be able to adapt it to find a path of desired length n. In doing so, you might even be able to achieve a couple of optimizations by 'short-circuiting' esp. when dealing with weighted edges or nodes—just don't visit any further paths where adding the weight of the edge or node would exceed your target.

If your desired length is small, then the color-coding method is pretty reasonable. See http://en.wikipedia.org/wiki/Color-coding. If k is the length of your desired path, the runtime of the algorithm is [math]O(k^k n)[/math]. The algorithm consists of many iterations where you randomly color all of the vertices with [math]k[/math] colors independently with colors 1, 2, ..., k. In the resulting colored graph, you then look for a path of length [math]k[/math] from your starting vertex [math]u[/math] to your terminal vertex [math]v[/math] that contains vertices colored with 1, 2, 3, ..., k. This can be found using a BFS that, when at a vertex with color i, only proceeds to vertices with color i + 1. With probability 1/k^k, a path of length k, if it exists, will be colored with colors 1, 2, ..., k in order. Therefore, you need around k^k iterations to discover the path with high probability. If your graph has many paths of this length between u and v, it will need fewer iterations to find the path.