On f-derangements and decomposing bipartite graphs into paths

Abstract

Let f : {1, . . . , n} → {1, . . . , n} be a function (not necessarily one-to-one). An fderangement is a permutation g : {1, . . . , n} → {1, . . . , n} such that g(i) ̸ = 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.