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This question was originally posted on mathoverflow. Having obtained no answers after one month, I've decided to cross-post it here.


Let $H = ( V, E )$ be a $k$-uniform connected hypergraph, with $n = |V|$ vertices and $m = |E|$ hyperedges. Let $O_w$ be the number of distinct edge induced subgraphs of $H$ having $w$ vertices and an odd number of hyperedges. Let $E_w$ be the number of distinct edge induced subgraphs of $H$ having $w$ vertices and an even number of hyperedges. Let $\Delta_w = O_w - E_w$.

Let $b_w$ be the number of bits required to encode $\Delta_w$, i.e. $b_w = log_2 \ \Delta_w$.

Let $b = \displaystyle\max_{\substack{w}} b_w$.

I'm interested in how $b$ grows. I would like to determine the best possible upper bound for $b$ which is expressible as a function of only $n$, $m$ and $k$. More precisely, I would like to determine a function $f(n,m,k)$ having both the following properties:

  1. $b \leq f( n, m, k )$ for any $k$-uniform hypergraph $H$ having $n$ vertices and $m$ hyperedges.
  2. $f(n,m,k)$ grows slower than any other function which satisfies the 1st property.

In general both $O_w$ and $E_w$ are exponential in $m$, therefore I expect that their difference $\Delta_w$ is not exponential in $m$ and thus that $b \in o( m )$.

However for the moment I've no clue on how to try to prove this.

Questions

  • How does $b$ grow with respect to $n$, $m$ and $k$?
  • Are there any relevant results in the literature?
  • Any hint on how to try to prove $b \in o(m)$?
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It is the case that $b \notin o( m )$.

Let $G = ( V_G, E_G )$ be a $3$-regular connected graph. Without loss of generality, let us fix $w = |V_G|$ and focus on $\Delta_{|V_G|} = O_{|V_G|} - E_{|V_G|}$.

Pick an arbitrary edge $e = \{u, v\} \in E_G$. Remove $e$ and replace it with the following gadget:

Gadget

This has the effect of multiplying $\Delta_{|V_G|}$ by $8$. The resulting graph $H = (V_H, E_H)$ is still $3$-regular connected, and has $60$ vertices more than $G$.

By repeating this process $i$ times, we obtain a $3$-regular connected graph having $|V_H| = |V_G| + 60i$ vertices, and whose $\Delta_{|V_H|}$ is $8^i$ times bigger than $\Delta_{|V_G|}$.

After $i = \frac{|V_G|}{3}$ iterations, our final graph will have $|V_H| = 21|V_G|$ vertices and a $\Delta_{|V_H|}$ which is $2^{|V_G|}$ times bigger than the $\Delta_{|V_G|}$ we started with. Let $n = |V_H|$, recall how $m = 1.5n$ for $3$-regular graphs, and we get a $\Delta_{|V_H|}$ whose absolute value is lower-bounded by $2^{\frac{m}{31.5}}$. Henceforth $b$ is linear in $m$ in the worst case, and thus it cannot belong to $o( m )$.

Contrarily to intuition (at least mine), the number of edge covers of odd cardinality and the number of edge covers of even cardinality may diverge by an exponential amount.

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