Vertex degree sums for perfect matchings in 3-uniform hypergraphs

Abstract

Let $n\equiv 0\left(\text{mod }3\right)$ and $H_{n,n/3}^2$ be the 3-graph of order n, whose vertex set is partitioned into two sets S and T of size $\frac{1}{3}n+1$ and $\frac{2}{3}n-1$, respectively, and whose edge set consists of all triples with at least 2 vertices in T. Suppose that n is sufficiently large and H is a 3-uniform hypergraph of order n with no isolated vertex. Zhang and Lu [Discrete Math. 341 (2018), 748–758] conjectured that if $\mathrm{deg}\left(u\right)+\mathrm{deg}\left(v\right)>2(\left(\begin{array}{c}n-1\\ 2\end{array}\right)-\left(\begin{array}{c}2n/3\\ 2\end{array}\right))$ for any two vertices u and v that are contained in some edge of H, then H contains a perfect matching or H is a subgraph of $H_{n,n/3}^2$. We construct a counter-example to the conjecture. Furthermore, for all $\gamma >0$ and $n\in 3Z$ sufficiently large, we prove that if $\mathrm{deg}\left(u\right)+\mathrm{deg}\left(v\right)>(3/5+\gamma )n^2$ for any two vertices u and v that are contained in some edge of H, then H contains a perfect matching or H is a subgraph of $H_{n,n/3}^2$. This implies a result of Zhang, Zhao and Lu [Electron. J. Combin. 25 (3), 2018].