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

Research output: Contribution to journalArticle

Colleges, School and Institutes

Abstract

In this paper we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [$1$-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, $\chi'(G)=D$. (ii) [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. (iii) [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 ${\rm reg}_{\rm even}(n,\delta)/2 \ge (n-2)/8$ edge-disjoint Hamilton cycles. Here $\text{reg}_{\text{even}}(n,\delta)$ 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.

Details

Original languageEnglish
Number of pages170
JournalMemoirs of the American Mathematical Society
Volume254
Issue number1154
Early online date21 Jun 2016
Publication statusE-pub ahead of print - 21 Jun 2016