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Proof of the $1$-factorization and Hamilton Decomposition Conjectures

About this Title

Béla Csaba, Bolyai Institute, University of Szeged, H-6720 Szeged, Aradi vértanúk tere 1. Hungary, Daniela Kühn, School of Mathematics University of Birmingham Edgbaston Birmingham B15 2TT UK, Allan Lo, School of Mathematics University of Birmingham Edgbaston Birmingham B15 2TT UK, Deryk Osthus, School of Mathematics University of Birmingham Edgbaston Birmingham B15 2TT UK and Andrew Treglown, School of Mathematics University of Birmingham Edgbaston Birmingham B15 2TT UK

Publication: Memoirs of the American Mathematical Society
Publication Year: 2016; Volume 244, Number 1154
ISBNs: 978-1-4704-2025-3 (print); 978-1-4704-3508-0 (online)
DOI: https://doi.org/10.1090/memo/1154
Published electronically: June 21, 2016
Keywords: 1-factorization, Hamilton cycle, Hamilton decomposition
MSC: Primary 05C70, 05C45

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Table of Contents

Chapters

  • 1. Introduction
  • 2. The two cliques case
  • 3. Exceptional systems for the two cliques case
  • 4. The bipartite case
  • 5. Approximate decompositions

Abstract

In this paper we prove the following results (via a unified approach) for all sufficiently large $n$:

  1. [$1$-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into perfect matchings. Equivalently, $\chi ’(G)=D$.

  2. [Hamilton decomposition conjecture] Suppose that $D \ge \lfloor n/2 \rfloor$. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into Hamilton cycles and at most one perfect matching.

  3. [Optimal packings of Hamilton cycles] Suppose that $G$ is a graph on $n$ vertices with minimum degree $\delta \ge n/2$. Then $G$ contains at least $\textrm {reg}_\textrm {even}(n,\delta )/2 \ge (n-2)/8$ edge-disjoint Hamilton cycles. Here $\mathrm {reg}_{\mathrm {even}}(n,\delta )$ denotes the degree of the largest even-regular spanning subgraph one can guarantee in a graph on $n$ vertices with minimum degree $\delta$.

(i) was first explicitly stated by Chetwynd and Hilton. (ii) and the special case $\delta = \lceil n/2 \rceil$ of (iii) answer questions of Nash-Williams from 1970. All of the above bounds are best possible.

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