Reconstructing hypergraph matching polynomials
Donggyu Kim, Hyunwoo Lee
TL;DR
The paper proves that the hypergraph matching polynomial $m_k(H,x)$ of an $n$-vertex $k$-graph is uniquely determined by the multiset of induced subhypergraphs on $n' = \left\lfloor \frac{k-1}{k}n\right\rfloor + 1$ vertices, extending Godsil’s reconstruction result from graphs to uniform hypergraphs. It achieves this via Lee’s hypergraph version of Godsil’s identity, constructing a digraph encoding $H$ (through $D(H,v)$) and linking $m_k(H,x)$ to the digraph’s characteristic polynomial, with a key identity relating derivatives of $m_k$ to closed walks in these digraphs. The authors provide explicit sharpness constructions (aligned-teeth and misaligned-teeth) showing the bound is tight, and derive a corollary for $F$-tiling polynomials, including the ability to count perfect $F$-tilings from the same subhypergraph data. Overall, the work extends reconstruction phenomena to hypergraphs, enabling recovery of spanning-structure counts from partial information and suggesting further directions for connected invariants like Hamilton cycles.
Abstract
By utilizing the recently developed hypergraph analogue of Godsil's identity by the second author, we prove that for all $n \geq k \geq 2$, one can reconstruct the matching polynomial of an $n$-vertex $k$-uniform hypergraph from the multiset of all induced sub-hypergraphs on $\lfloor \frac{k-1}{k}n \rfloor + 1$ vertices. This generalizes the well-known result of Godsil on graphs in 1981 to every uniform hypergraph. As a corollary, we show that for every graph $F$, one can reconstruct the number of $F$-factors in a graph under analogous conditions. We also constructed examples that imply the number $\lfloor \frac{k-1}{k}n \rfloor + 1$ is the best possible for all $n\geq k \geq 2$ with $n$ divisible by $k$.