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A note on colour-bias perfect matchings in hypergraphs

József Balogh, Andrew Treglown, Camila Zárate-Guerén

Abstract

A result of Balogh, Csaba, Jing and Pluhár yields the minimum degree threshold that ensures a $2$-coloured graph contains a perfect matching of significant colour-bias (i.e., a perfect matching that contains significantly more than half of its edges in one colour). In this note we prove an analogous result for perfect matchings in $k$-uniform hypergraphs. More precisely, for each $2\leq \ell <k$ and $r\geq 2$ we determine the minimum $\ell$-degree threshold for forcing a perfect matching of significant colour-bias in an $r$-coloured $k$-uniform hypergraph.

A note on colour-bias perfect matchings in hypergraphs

Abstract

A result of Balogh, Csaba, Jing and Pluhár yields the minimum degree threshold that ensures a -coloured graph contains a perfect matching of significant colour-bias (i.e., a perfect matching that contains significantly more than half of its edges in one colour). In this note we prove an analogous result for perfect matchings in -uniform hypergraphs. More precisely, for each and we determine the minimum -degree threshold for forcing a perfect matching of significant colour-bias in an -coloured -uniform hypergraph.
Paper Structure (6 sections, 6 theorems, 11 equations, 1 figure)

This paper contains 6 sections, 6 theorems, 11 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and useful results
  3. Proof overview and key definitions
  4. Tools for the proof of Theorem \ref{['thmmain2']}
  5. Proof of Theorem \ref{['thmmain2']}
  6. Concluding Remarks

Key Result

Theorem 1.1

Let $0<c<1/4$ and $n\in \mathbb N$ be sufficiently large. If $G$ is an $n$-vertex graph with then given any $2$-colouring of $E(G)$ there is a Hamilton cycle in $G$ with at least $n/2+cn/32$ edges of the same colour. Moreover, if $n\in 4 \mathbb N$, there is an $n$-vertex graph $G'$ with $\delta(G')=3n/4$ and a $2$-colouring of $E(G')$ for which every Hamilton cycle in $G'$ has precisely $n

Figures (1)

  • Figure 1: On the left, a $(12, e, f)$-gadget. On the right, a $(9,e,f)$-gadget.

Theorems & Definitions (15)

  • Theorem 1.1: Balogh, Csaba, Jing and Pluhár BCsJP
  • Theorem 1.2: Balogh, Csaba, Jing and Pluhár BCsJP
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 5 more