Given an undirected graph with an edge set $E$, and where an edge is an unordered pair of vertices, how many ordered sequences of size $L$ of edges from $E$ exist such that all the vertices that are present in a sequence appear an even number of times?
For example $\{(x,y),(z,w),(x,y),(z,w)\}$ is a sequence of size $L=4$. The edges participating in the sequence are $(x,y)$ and $(z,w)$ and each of the vertices $x,y,z,w$ appears exactly twice.
I am particularly interested in edge sets of planar graphs (e.g., two-dimensional square or triangular lattices).