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Vertex degree sums for perfect matchings in 3-uniform hypergraphs

Yan Wang, Yi Zhang

Abstract

Let $n \equiv 0\, (\, \text{mod } 3\,)$ 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 $deg(u)+deg(v) > 2(\binom{n-1}{2}-\binom{2n/3}{2})$ 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 $H_{n,n/3}^2$. We construct a counter-example to the conjecture. Furthermore, for all $γ>0$ and let $n \in 3 \mathbb{Z}$ be sufficiently large, we prove that if $deg(u)+deg(v) > (3/5+γ)n^2$ 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].

Vertex degree sums for perfect matchings in 3-uniform hypergraphs

Abstract

Let and be the 3-graph of order , whose vertex set is partitioned into two sets and of size and , respectively, and whose edge set consists of all triples with at least vertices in . Suppose that is sufficiently large and is a 3-uniform hypergraph of order with no isolated vertex. Zhang and Lu [Discrete Math. 341 (2018), 748--758] conjectured that if for any two vertices and that are contained in some edge of , then contains a perfect matching or is a subgraph of . We construct a counter-example to the conjecture. Furthermore, for all and let be sufficiently large, we prove that if for any two vertices and that are contained in some edge of , then contains a perfect matching or is a subgraph of . This implies a result of Zhang, Zhao and Lu [Electron. J. Combin. 25 (3), 2018].
Paper Structure (6 sections, 19 theorems, 79 equations, 3 figures)

This paper contains 6 sections, 19 theorems, 79 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thextremegraph']}
  3. Outline of the proof of Theorem \ref{['Yi1']} and preliminaries
  4. Absorbing lemma \ref{['absorbinglemma1']}
  5. Proof of Lemma \ref{['lemma11181']}
  6. Proof of Lemma \ref{['lemma11182']}

Key Result

Theorem 1

Kuhn2 There exists $n_0 \in \mathbb{N}$ such that if $H$ is a $3$-graph of order $n \geq n_0$ with $\delta_1(H) > \delta_1(H^1_{n, s})= \binom{n-1}{2}-\binom{n-s}{2}$ and $n \geq 3s$, then $H$ contains a matching of size $s$.

Figures (3)

  • Figure 1: The graph $H^{1,2}_{n,x,y}$.
  • Figure 2: The graphs $B_{113}$, $B_{023}$ and $B_{033}$.
  • Figure 3: An absorbing $2k^2$-sets $T$ for $A$ ($T$ is red and$H[A \cup T]$ is green).

Theorems & Definitions (47)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Conjecture 4
  • Theorem 5
  • Theorem 6
  • Lemma 7
  • proof
  • Conjecture 8
  • Lemma 9
  • ...and 37 more