Abstract
We determine the minimum degree sum of two adjacent vertices that ensures a perfect matching in a 3-uniform hypergraph without an isolated vertex. Suppose that $H$ is a 3-uniform hypergraph whose order $n$ is sufficiently large and divisible by $3$. If $H$ contains no isolated vertex and $\deg(u)+\deg(v) > \frac{2}{3}n^2-\frac{8}{3}n+2$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $H$ contains a perfect matching. This bound is tight and the (unique) extremal hyergraph is a different space barrier from the one for the corresponding Dirac problem.
Highlights
A k-uniform hypergraph H is a pair (V, E), where V := V (H) is a finite set of vertices and E := E(H) is a family of k-element subsets of V
Given integers l < k ≤ n such that k divides n, we define the minimum l-degree threshold ml(k, n) as the smallest integer m such that every k-graph H on n vertices with δl(H) ≥ m contains a perfect matching
We determine the largest σ2′ (H) among all 3-graphs H of order n without isolated vertex such that H contains no perfect matching. (Trivially H contains no perfect matching if it contains an isolated vertex.) Let us define a 3-graph H∗, whose vertex set is partitioned into two vertex classes S and T of size n/3 + 1 and 2n/3 − 1, respectively, and whose edge set consists of all the triples containing at least two vertices of T
Summary
A k-uniform hypergraph (in short, k-graph) H is a pair (V, E), where V := V (H) is a finite set of vertices and E := E(H) is a family of k-element subsets of V. Given integers l < k ≤ n such that k divides n, we define the minimum l-degree threshold ml(k, n) as the smallest integer m such that every k-graph H on n vertices with δl(H) ≥ m contains a perfect matching. We determine the largest σ2′ (H) among all 3-graphs H of order n without isolated vertex such that H contains no perfect matching. (Trivially H contains no perfect matching if it contains an isolated vertex.) Let us define a 3-graph H∗, whose vertex set is partitioned into two vertex classes S and T of size n/3 + 1 and 2n/3 − 1, respectively, and whose edge set consists of all the triples containing at least two vertices of T. More precisely there are increasing functions f and g such that given a3, whenever we choose some a2 ≤ f (a3) and a1 ≤ g(a2), all calculations needed in our proof are valid
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