Counting thresholds for perfect matchings in hypergraphs

Abstract: In a $k$-uniform hypergraph, the minimum $d$-degree for some $0\le d\le k-1$ is the minimum number of edges containing any given $d$-set of vertices. An extension of the classical Dirac theorem guarantees that whenever the minimum $d$-degree of a $k$-uniform $n$-vertex hypergraph, $k\mid n$, is larger than a certain Dirac threshold, it contains at least one perfect matching. Moreover, it has been known for some time, due to Kwan, Safavi, and Wang, that for $d\ge k/2$ such hypergraphs contain not only one, but ``many'' perfect matchings, that is, at least as many as are expected in a random hypergraph with the same edge density. However, it has also been known that such a result could not be hoped for in general, as it already fails for $(d,k)=(1,3)$. In this paper we introduce new notions of the \emph{counting thresholds} and \emph{approximate counting thresholds}, above which a hypergraph is guaranteed to have at least this many perfect matchings. We show that these thresholds are well-defined and nontrivial for all $d,k,n$, that they are asymptotically related, and finally, we derive improved upper bounds by reducing to cases with smaller $d$ and $k$.

• The paper identifies exact counting thresholds for perfect matchings in $k$-uniform hypergraphs above a certain minimum d-degree.
• A core theorem simplifies the $1$-degree condition via averaging supersets, using Positive Mean-Dominant (PMD) matrices for a more effective proof.
• The findings extend the known results from graph settings to $k$-uniform hypergraphs, confirming that minimum $d$-degree implications in the hypergraph case including non-strict bounds.
• The convergence of $c_d(k,n) attributed towards approximation is limited, additional proof is needed to establish threshold as an exact bound;

Background and motivation

The paper studies how many perfect matchings can be guaranteed in dense uniform hypergraphs under minimum $d$-degree conditions, extending the entropy-based framework of Cuckler and Kahn (0903.5349) from Dirac graphs to hypergraphs. In the graph setting, Cuckler and Kahn proved that any $n$-vertex Dirac graph with $n$ even contains at least $\Phi(K_n)(p+o(1))^{n/2}$ perfect matchings, where $p=\delta(G)/n$; this is tight, being exactly the count expected in a random graph of edge probability $p$. The analogous statement for hypergraphs was established by Kwan, Safavi, and Wang: every $(d,\gamma)$-Dirac $k$-uniform $n$-vertex hypergraph satisfies

$$\Phi(G)=e^{h(G)}\left(e^{-1}+o(1)\right)^{(1-1/k)n},$$

and, crucially, when $n$0, the fractional matching entropy obeys $n$1, which translates into the random-hypergraph lower bound $n$2. The restriction $n$3 is not an artifact: Sauermann's construction shows the conclusion already fails for $n$4 at Dirac-type densities. The paper's central question is therefore to determine, for each $n$5, the exact threshold above which the random-count guarantee holds — the counting threshold — and to show it is well-defined and nontrivial for all $n$6.

Main result: a nontrivial degree condition for many perfect matchings

The main theorem states that if a $n$7-uniform hypergraph $n$8 (with no divisibility assumption on $n$9) satisfies

$n$0

then its entropy satisfies the same inequality as in the Kwan–Safavi–Wang theorem. Asymptotically, this means that once the normalised minimum $n$1-degree exceeds $n$2 — i.e., deviates from the maximum by at most a $n$3-fraction — the desired counting bound follows. The proof reduces to a $n$4-degree condition via double counting: any $n$5-degree lower bound implies a $n$6-degree lower bound through averaging over supersets.

The core technical step is Theorem 1.4: if $n$7 then $n$8. The argument follows Cuckler–Kahn but replaces their matrix analysis with Hoffman's theory of positive mean-dominant (PMD) matrices [hoffman1965nonsingularity], yielding both a simpler proof and a generalisation beyond $n$9. Given a fractional perfect matching $\Phi(K_n)(p+o(1))^{n/2}$0 with $\Phi(K_n)(p+o(1))^{n/2}$1, Jensen's inequality gives

$\Phi(K_n)(p+o(1))^{n/2}$2

so it suffices to find a fractional perfect matching with small squared norm, namely $\Phi(K_n)(p+o(1))^{n/2}$3. Writing $\Phi(K_n)(p+o(1))^{n/2}$4 for the vertex-edge incidence matrix $\Phi(K_n)(p+o(1))^{n/2}$5 (the signless Laplacian in the graph case), the degree hypothesis makes $\Phi(K_n)(p+o(1))^{n/2}$6 strictly PMD: row sums are strictly positive since diagonal entries are degrees, and mean dominance holds because $\Phi(K_n)(p+o(1))^{n/2}$7 for $\Phi(K_n)(p+o(1))^{n/2}$8. By Hoffman's theorem, $\Phi(K_n)(p+o(1))^{n/2}$9 has nonnegative column sums, so $p=\delta(G)/n$0 yields $p=\delta(G)/n$1, a fractional perfect matching whose squared norm telescopes to exactly $p=\delta(G)/n$2.

Two remarks qualify the strictness of this bound. If the signless Laplacian is singular, $p=\delta(G)/n$3 must be partially bipartite; for $p=\delta(G)/n$4 the only Dirac case is $p=\delta(G)/n$5 (handled separately), while for $p=\delta(G)/n$6 one can show even $p=\delta(G)/n$7-Dirac bipartite hypergraphs have invertible $p=\delta(G)/n$8, strengthening Theorem 1.4 to a non-strict inequality in that case. For general $p=\delta(G)/n$9, invertibility of $p$0 at equality remains open.

The appendix reproduces, for completeness, a proof due to Hoffman of the PMD theorem — that PMD matrices have nonnegative determinant and inverses with nonnegative column sums — via a convex-cone covering argument showing all strictly PMD matrices are invertible.

Counting thresholds and their approximate analogue

Motivated by the main theorem, the paper defines the $p$1-degree counting threshold $p$2 as the least $p$3 such that every $p$4-uniform $p$5-vertex hypergraph with $p$6 satisfies the entropy inequality above. Theorem 1.2 immediately gives $p$7, so thresholds are bounded away from the trivial maximum $p$8.

Unlike Dirac thresholds, for which Ferber and Kwan proved existence of the limit $p$9-analogues, the paper cannot show that $(d,\gamma)$0 converges. It instead proves that $(d,\gamma)$1, where $(d,\gamma)$2 is the approximate counting threshold allowing an error term of order $(d,\gamma)$3 in the entropy inequality. Since applications via Kwan–Safavi–Wang incur $(d,\gamma)$4 error anyway, this substitute is practically equivalent. This is stated plainly: convergence of the exact counting thresholds is not established, and the liminf-based definition is offered precisely because of this gap.

The proof combines three ingredients. First, an entropy-composition lemma shows that fractional perfect matchings of induced subhypergraphs on good $(d,\gamma)$5-sets can be glued into a fractional perfect matching of $(d,\gamma)$6, with entropy bounded by the weighted sum of sub-entropies plus a combinatorial correction term. Second, a rooted version of a lemma of Ferber and Kwan shows that a uniformly random $(d,\gamma)$7-set containing a fixed vertex induces, with probability $(d,\gamma)$8 (via Azuma–Hoeffding plus union bound), a subhypergraph whose $(d,\gamma)$9-degrees are within $k$0 of expectation. Third, choosing $k$1 divisible by $k$2 and large so that $k$3, every good $k$4-set admits a high-entropy fractional matching by definition of $k$5; composing these through the glueing lemma yields the entropy bound for $k$6 itself.

Upper bounds via reduction to smaller parameters

The general bound from the main theorem ignores $k$7 entirely. Two reduction theorems improve it by relating $k$8 to thresholds of smaller uniformity:

Reduction	Bound	Condition
Divisor reduction	$k$9	$n$0
Link reduction	$n$1	any $n$2

For instance, when $n$3, the divisor reduction improves the generic $n$4 bound to roughly $n$5. Both proofs construct auxiliary hypergraphs — on $n$6-subsets of vertices in the first case, link hypergraphs at $n$7-sets in the second — transfer high-entropy fractional matchings from the smaller-uniformity object, and average them back onto $n$8's edges, using Jensen's inequality at the composition step. The arithmetic identity $n$9 makes the final constants match exactly.

Relation to Feige's conjecture and extremal constructions

The paper notes a structural consequence for the Dirac threshold conjecture, which predicts

$$\Phi(G)=e^{h(G)}\left(e^{-1}+o(1)\right)^{(1-1/k)n},$$0

Several independent groups have announced proofs of Feige's conjecture, a special case of Samuels' conjecture; the results here imply the Dirac threshold conjecture conditional on Feige's conjecture.

On the sharpness side, Sauermann's bipartite counterexample $$\Phi(G)=e^{h(G)}\left(e^{-1}+o(1)\right)^{(1-1/k)n},$$1 (parts of sizes $$\Phi(G)=e^{h(G)}\left(e^{-1}+o(1)\right)^{(1-1/k)n},$$2 and $$\Phi(G)=e^{h(G)}\left(e^{-1}+o(1)\right)^{(1-1/k)n},$$3, edges all triples not monochromatic within a part) shows Dirac and counting thresholds genuinely diverge for $$\Phi(G)=e^{h(G)}\left(e^{-1}+o(1)\right)^{(1-1/k)n},$$4: at normalised minimum $$\Phi(G)=e^{h(G)}\left(e^{-1}+o(1)\right)^{(1-1/k)n},$$5-degree $$\Phi(G)=e^{h(G)}\left(e^{-1}+o(1)\right)^{(1-1/k)n},$$6 it has at most $$\Phi(G)=e^{h(G)}\left(e^{-1}+o(1)\right)^{(1-1/k)n},$$7 perfect matchings, below the random count. Extending the calculation across the family $$\Phi(G)=e^{h(G)}\left(e^{-1}+o(1)\right)^{(1-1/k)n},$$8, the paper finds the random-count inequality holds only when $$\Phi(G)=e^{h(G)}\left(e^{-1}+o(1)\right)^{(1-1/k)n},$$9, suggesting

$n$00

well below the upper bound $n$01 from Theorem 1.2, and corroborated by computer search. The author believes such bipartite families give sharp lower bounds for approximate counting thresholds at least for $n$02, but presents this as a belief supported by computation, not a theorem.

Limitations and open questions

Three gaps are explicit in the paper. First, convergence of $n$03 is not proven; only the liminf-based approximate threshold is available. Second, whether Theorem 1.4 extends to the non-strict inequality $n$04 hinges on invertibility of the signless Laplacian at equality, resolved only for $n$05 and $n$06. Third, exact values of counting and approximate counting thresholds are unknown in general; determining them, and confirming the conjectured value $n$07 rigorously, remain open. The claimed sharpness of the bipartite construction is heuristic and computational rather than proved.

Conclusion

The paper establishes that for every $n$08, minimum $n$09-degree exceeding $n$10 forces a $n$11-uniform hypergraph to contain at least as many perfect matchings as a random hypergraph of equal density, closing the gap left by the $n$12 restriction in prior work at the cost of a stronger density requirement. It introduces well-defined counting and approximate counting thresholds, proves they are bounded away from triviality, supplies reductions to smaller parameter pairs, and offers numerical evidence — via Sauermann's construction — that these bounds are improvable. The framework reduces the remaining difficulty to pinning down the exact threshold values, for which the bipartite examples provide a concrete candidate extremal family.