Coarse-graining a bounded-incidence hypergraph

Question

Let $G$ be a connected hypergraph with vertex-set $V$ and hyperedge-set $E \subseteq 2^V\smallsetminus \{\emptyset\}$. Assume that every vertex is part of at most $\Delta$ hyperedges and every hyperedge has at most $\Delta$ vertices. I call such $G$ a bounded-incidence hypergraph.

Let $n\ge1$ be an integer and $c>0$ be a real constant. I define a good cover of $G$ as a collection $\{G_1,G_2,\ldots\}$ of sub-hypergraphs of $G$ such that

1. each $G_i$ is a connected, induced sub-hypergraph of $G$,
2. there is an integer $m\ge1$ such that every hyperedge of $G$ is contained in at least one and at most $m$ distinct $G_i$'s.
3. there is an integer $N\ge n$ such that $n\le |V(G_i)|\le N$ for each $i$, and
4. for each $i$, we have $$ \sum_{j\,:\, G_i \cap G_j \ne \emptyset} e^{-c |V(G_i \cap G_j)|} \le \frac12. $$

Given an individual hypergraph $G$, it is easy to construct good covers, but I want to do this uniformly for all bounded-incidence hypergraphs. More precisely,

Question: are there integers $m$ and $N$ which depend only on $\Delta$, $n$, and $c$ such that every hypergraph of bounded-incidence $\Delta$ has a good cover with these parameters for sufficiently large $n$?

Edit: As per @caduk's comment, here is a more precise version of the question. Is the following statement true?

$\forall \Delta,c, \exists n_0, \forall n>n_0, \exists m,N, \forall G$ with $|V(G)| \ge n$ and maximum incidence $\Delta$, there is a good cover of $G$ with these parameters.

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One can interpret $G_i$'s as "coarse-grained" hyperedges that form a "coarse-grained" hypergraph on the original vertex-set $V$. Intuitively, the coarse-grained hyperedges contain at least $n$ vertices, but not too many due to condition 3. More importantly, the number of vertices in the overlap between two coarse-grained hyperedges cannot be too small due to condition 4. At the same time, the number of such overlaps (i.e., the number of coarse-grained hyperedges containing a given vertex) cannot be too large due to condition 2.

While these may seem impossible to be satisfied simultaneously, it is easy to construct a good cover uniformly for all graphs (i.e., each hyperedge is an edge). The idea is to construct a good cover uniformly for all trees and then use the fact that every graph has a spanning tree. I know an algorithmic way to construct a good cover of any tree such that $m \sim \Delta^3 n$ and $N \sim \Delta^3 n^2$. (See this answer to a related question I posted on mathoverflow a while ago.) However, for hypergraphs, there is no spanning tree in general, so this algorithm does not extend easily.

I would appreciate any help in solving this problem, or pointing me to any relevant literature.

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Update: I realized that the fourth condition can be satisfied if the following conditions are met:

• for each $i,j$, either $G_i \cap G_j = \emptyset$ or $|V(G_i\cap G_j)| \ge n$, and
• $N$ and $m$ grow polynomially in $n$.

Let me explain why these are sufficient. First, for each $i$, let us count the number of $j$ such that $G_i \cap G_j \ne \emptyset$. Since each $G_i$ has at most $N$ vertices, each vertex is part of at most $\Delta$ hyperedges, and each hyperedge is contained in at most $m$ $G_j$'s, there are at most $N \times \Delta \times m$ $G_j$'s that intersect $G_i$ nontrivially. Therefore, the fourth condition is satisfied because $$ \sum_{j\,:\,G_i \cap G_j \ne \emptyset} e^{-c|V(G_i\cap G_j)|} \le \Delta N m e^{-cn}, $$ which can be made smaller than $1/2$ for sufficiently large $n$.

Both these conditions are indeed satisfied in the case of graphs. So the question is if this is the case for hypergraphs too. I would appreciate any thoughts or comments.

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Update 2:

I wanted to explain quickly how to prove that a good cover exists for all graphs. I first want to construct a partition and then construct a good cover using it. Let $P$ be the set of "covered" vertices, which is initially empty. In the induced subgraph on $V(G)\setminus P$, pick a largest possible connected induced subgraph of order at most $n+1$. If the order of this subgraph is at least $n$, call it a "good" part, else call it "bad" part. Add the vertices in this part to $P$ and repeat until $P=V(G)$. The nice thing about this construction is that the "bad" parts are never adjacent to each other (if they were adjacent, we could have built a larger connected part by combining them). Now, define the $G_i$'s for the good cover as follows: for each part, take the union of that part and all the "good" parts adjacent to it. It is easy to see that these $G_i$'s satisfy conditions 1, 2, and 3. For condition 4, I use the formulation given in the update above. Indeed, if $G_i$ and $G_j$ intersect, they must do so on a "good" part because no two "bad" parts are adjacent. Therefore, any nontrivial intersection contains at least $n$ vertices. QED.

The key issue with extending this approach to hypergraphs is that the "bad" parts could be adjacent. So the argument for "intersection of $\ge n$" doesn't go through. I don't know how to avoid this. The existing answer below has a very similar problem. It satisfies the first three conditions, but not the "intersection of $\ge n$" condition.

Answer 1

I think the following should work:

We will create a partition of the vertices in connected induced components of size $n+\Delta$ or less which we will call parts. The parts are pairwise disjoint. A part of size $n$ or more is said complete. Each $G_i$ will contain at least one complete part. If two $G_i$ overlap, they overlap on at least one complete part.

We build the parts sequentially. Suppose we are building the $i$-th part. Let $P$ be the union of all the parts created before this step.

Let $v\in V(G)-P$ be a vertex not yet included in a part. We create the biggest possible part $p_i$ by taking a connected induced graph of $G-P$ containing $v$. We repeat until the parts partition $V(G)$.

For each part $p_i$, we create $G_i$ by taking $p_i$, all the edges that overlap $p_i$, and all the complete parts that overlap some of these edges. Note that all edges are now covered.

We verify that we satisfy the requirements:

We set:

• $N=(n+\Delta)+(n+\Delta)\Delta(\Delta-1)(n+\Delta)$

  $G_i$ contains the part $p_i$ (size bounded by $n+\Delta$), and for each vertex $v$ of $p_i$ ($n+\Delta$ possible), for each edge $e$ containing $v$ ($\Delta$ possible), we add $e$, and for each $v'\in e, v'\neq v$ ($\Delta-1$ possible) $G_i$ can contain the complete part containing $v'$ (bounded by $n+\Delta$)

• $m=1+(n+\Delta)(\Delta-1)(\Delta-1)$

  If an edge $e$ is totally included in a part, it must be in $p_i$ for some $G_i$. The edge $e$ belongs to $G_i$, and if $p_i$ is complete, for each vertex $v$ of $p_i$ ($n+\Delta$), for each edge $e'$ that contains $v$ (but not contained in the part) ($\Delta-1$), for each vertex of $v'\in e', v'\neq v$ ($\Delta-1$), if $p_j$ is the part containing $v'$, $G_j$ contains $e$. If the edge is not included in a part, it belongs to at most $\Delta$ subgraphs (once for each vertex of the edge).

Every edge $e$ overlapping at least two parts must overlap at least one complete part because we could otherwise add $e$ to one of these parts to get a bigger part. For every non-complete part, a partially overlapping edge must exist (the graph is connected and of size at least $n$), so every $G_i$ will contain at least one complete part, and $|G_i|>n$.

If two subgraph overlap on a complete part, they both contain this complete part. Otherwise, they must overlap on a vertex covered by an edge $e$ that overlap multiple parts. Then they both contain at least one complete part spanned by $e$, thus their intersection is of size at least $n$.