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Rainbow perfect matchings in 3-partite 3-uniform hypergraphs

Hongliang Lu, Yan Wang

TL;DR

This work resolves rainbow perfect matching thresholds for families of balanced $3$-partite $3$-graphs: for $m=3r+s$ with $1\le s\le 3$, define the piecewise threshold $\delta(n,r,s)$ and show there exists $n_0$ such that if each $F_i$ has $n$ vertices per class and minimum vertex degree $\delta_1(F_i)\ge\delta(n,r,s)+1$, then the collection $\{F_1,\dots,F_n\}$ contains a rainbow perfect matching. The authors extend the single-graph results of Lo and Markström by leveraging fractional rainbow matching theory (Aharoni et al.) to obtain edge-disjoint fractional perfect matchings, and then deploy a combination of extremal (stability) analysis, an absorbing method, and nibble-based near-complete constructions to upgrade to an actual rainbow perfect matching. The approach provides a unified framework for rainbow matchings in $3$-partite $3$-graphs, with tight lower bounds demonstrated by extremal examples. The results have implications for multi-graph matching problems under shared vertex partitions and contribute to the broader program of rainbow and fractional matching techniques in hypergraphs.

Abstract

Let $m,n,r,s$ be nonnegative integers such that $n\ge m=3r+s$ and $1\leq s\leq 3$. Let \[δ(n,r,s)=\left\{\begin{array}{ll} n^2-(n-r)^2 &\text{if}\ s=1 , \\[5pt] n^2-(n-r+1)(n-r-1) &\text{if}\ s=2,\\[5pt] n^2 - (n-r)(n-r-1) &\text{if}\ s=3. \end{array}\right.\] We show that there exists a constant $n_0 > 0$ such that if $F_1,\ldots, F_n$ are 3-partite 3-graphs with $n\ge n_0$ vertices in each partition class and minimum vertex degree of $F_i$ is at least $δ(n,r,s)+1$ for $i \in [n]$ then $\{F_1,\ldots,F_n\}$ admits a rainbow perfect matching. This generalizes a result of Lo and Markström on the vertex degree threshold for the existence of perfect matchings in 3-partite 3-graphs. In this proof, we use a fractional rainbow matching theory obtained by Aharoni et al. to find edge-disjoint fractional perfect matching.

Rainbow perfect matchings in 3-partite 3-uniform hypergraphs

TL;DR

This work resolves rainbow perfect matching thresholds for families of balanced -partite -graphs: for with , define the piecewise threshold and show there exists such that if each has vertices per class and minimum vertex degree , then the collection contains a rainbow perfect matching. The authors extend the single-graph results of Lo and Markström by leveraging fractional rainbow matching theory (Aharoni et al.) to obtain edge-disjoint fractional perfect matchings, and then deploy a combination of extremal (stability) analysis, an absorbing method, and nibble-based near-complete constructions to upgrade to an actual rainbow perfect matching. The approach provides a unified framework for rainbow matchings in -partite -graphs, with tight lower bounds demonstrated by extremal examples. The results have implications for multi-graph matching problems under shared vertex partitions and contribute to the broader program of rainbow and fractional matching techniques in hypergraphs.

Abstract

Let be nonnegative integers such that and . Let \[δ(n,r,s)=\left\{\begin{array}{ll} n^2-(n-r)^2 &\text{if}\ s=1 , \\[5pt] n^2-(n-r+1)(n-r-1) &\text{if}\ s=2,\\[5pt] n^2 - (n-r)(n-r-1) &\text{if}\ s=3. \end{array}\right.\] We show that there exists a constant such that if are 3-partite 3-graphs with vertices in each partition class and minimum vertex degree of is at least for then admits a rainbow perfect matching. This generalizes a result of Lo and Markström on the vertex degree threshold for the existence of perfect matchings in 3-partite 3-graphs. In this proof, we use a fractional rainbow matching theory obtained by Aharoni et al. to find edge-disjoint fractional perfect matching.
Paper Structure (6 sections, 19 theorems, 50 equations)

This paper contains 6 sections, 19 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Small rainbow matchings
  3. Extremal case
  4. Absorbing lemma
  5. Almost perfect matchings in non-extremal case
  6. Proof of Theorem \ref{['main']}

Key Result

Theorem 1.1

Let $m,n,k$ be nonnegative integers with $k\ge 2$ and $n\ge k^7m$, and let $H$ be a $k$-partite $k$-graph with $n$ vertices in each class. If $\nu(H)= m$ and then $H$ is a subgraph of some member of ${\cal H}_k(n;m;\lceil m/k\rceil)\cup {\cal H}_k^*(n;m)$. Moreover, if $m\ne 1 \pmod k$ then $H$ is a subgraph of a member of ${\cal H}_k(n;m;\lceil m/k\rceil)$.

Theorems & Definitions (19)

  • Theorem 1.1: Lo and Markström LM
  • Theorem 1.2: Lo and Markström LM
  • Theorem 1.3: Lu and Zhang LZ17
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Theorem 3.2
  • Lemma 4.1: Lo and Markström, LM
  • ...and 9 more