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arXiv · 2201.02332

On f-Derangements and Decomposing Bipartite Graphs into Paths

Michael Plantholt·Hamidreza Habibi·Benjamin Mussell
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Abstract

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

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Michael Plantholt, Hamidreza Habibi, Benjamin Mussell. 2022-01-07. On f-Derangements and Decomposing Bipartite Graphs into Paths. https://arxiv.org/abs/2201.02332

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