On f-Derangements and Decomposing Bipartite Graphs into Paths

Let $f: \{1, ..., n\} \rightarrow \{1, ..., n\}$ be a function (not necessarily one-to-one). An $f-derangement$ is a permutation $g:\{1,...,n\} \rightarrow \{1,...,n\}$ such that $g(i) \neq f(i)$ for each $i = 1, ..., n$. When $f$ is itself a permutation, this is a standard derangement. We examine properties of f-derangements, and show that when we fix the maximum number of preimages for any item under $f$, the fraction of permutations that are f-derangements tends to $1/e$ for large $n$, regardless of the choice of $f$. We then use this result to analyze a heuristic method to decompose bipartite graphs into paths of length 5