The Elser Nuclei Sum Revisited

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Date
2021-03-16MFO Scientific Program
OWLF 2020Series
Oberwolfach Preprints;2021-05Author
Grinberg, Darij
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Show full item recordOWP-2021-05
Abstract
Fix a finite undirected graph $\Gamma$ and a vertex $v$ of $\Gamma$. Let $E$ be the set of edges of $\Gamma$. We call a subset $F$ of $E$ $\textit{pandemic}$ if each edge of $\Gamma$ has at least one endpoint that can be connected to $v$ by an $F$-path (i.e., a path using edges from $F$ only). In 1984, Elser showed that the sum of $\left(-1\right)^{\left| F\right|}$ over all pandemic subsets $F$ of $E$ is $0$ if $E\neq\varnothing$. We give a simple proof of this result via a sign-reversing involution, and discuss variants, generalizations and a refinement using discrete Morse theory.