#StackBounty: #boolean-functions #matching #monotone Perfect matching of monotone Boolean function with null Euler characteristic

Bounty: 100

For a set $V = {0,ldots,k}$ of variables, let $mathbf{G}_V$ be the undirected graph with set of vertices ${S subseteq V}$ and set of edges ${{S,S’} mid S subseteq S’ text{ and }|S’| = |S|+1}$, in other words the underlying undirected graph of the Hasse diagram of the powerset of $V$. Let $phi$ be a Boolean function on variables $V$, that we see as a set of subsets of $V$. Define the graph $mathbf{G}_V[phi]$ to be the subgraph of $mathbf{G}_V$ induced by $phi$.

I conjecture the following:

If $phi$ is monotone and $sum_{S in phi} (-1)^{|S|} = 0$, i.e., $phi$ has as many satisfying valuations of even and odd size, then $mathbf{G}_V[phi]$ or $mathbf{G}_V[lnotphi]$ has a perfect matching.

What I know:

  • For a Boolean function $phi$, $sum_{S in phi} (-1)^{|S|}$ is called the Euler characteristic of $phi$.
  • Observe that $sum_{S in phi} (-1)^{|S|} = – sum_{S in lnotphi} (-1)^{|S|}$ ($=0$ in our case).
  • Of course, if $sum_{S in phi} (-1)^{|S|} neq 0$, then neither $mathbf{G}_V[phi]$ nor $mathbf{G}_V[lnotphi]$ can have a perfect matching.
  • My claim does not hold if we do not impose $phi$ to be monotone. For example, consider the following Boolean function (the satisfying valuations of $phi$ are colored):not monotone
    $sum_{S in phi} (-1)^{|S|} = 0$, but $mathbf{G}_V[phi]$ does not have a perfect matching since, for instance, ${3,4}$ is isolated. The same holds for $mathbf{G}_V[lnotphi]$ by considering ${0,3,4}$.

  • There are examples where both have a perfect matching, and examples where only one does. Here is one such example (this is actually the smallest such example):enter image description here
    $sum_{S in phi} (-1)^{|S|} = 0$, $mathbf{G}_V[phi]$ does not have a perfect matching, but $mathbf{G}_V[lnotphi]$ does (trust me).
    I suspect that, if the claim is true, what decides if $mathbf{G}_V[phi]$ or $mathbf{G}_V[lnotphi]$ has a perfect matching is $max(|phi|,|lnotphi|)$.

  • There are no counterexamples to my claim for $k leq 5$.

  • In the particuliar case where $lnot phi$ corresponds to a collapsible abstract simplicial complex (an abstract simplicial complex is just another word for the negation of a monotone Boolean function), then $mathbf{G}_V[lnotphi]$ has a perfect matching.

What I have tried:

Since $mathbf{G}_V[phi]$ and $mathbf{G}_V[lnotphi]$ are bipartite, I’ve tried using:

  • Kőnig’s theorem. So here I would need to show that for $mathbf{G}_V[phi]$ (or $mathbf{G}_V[lnotphi]$), there is no vertex cover of size less that $|phi|/2$ (since the set of all subsets in $phi$ of even size is obviously a vertex cover of $mathbf{G}_V[phi]$).
  • Hall’s mariage theorem. Here I would need to show that, for every subset $B$ of $phi$ containing only subsets of even size, the subsets of $B$ are (collectively) adgacent to at least $|B|$ vertices in $mathbf{G}_V[phi]$.

with no luck so far. I should add that I don’t have any particular reason for why this should be true, but this came to me as an intriguing question while I was working on something else.

Get this bounty!!!

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