Proof of the 1-factorization and Hamilton Decomposition Conjectures

Ebook Details

Authors

Bela Csaba, Daniela Kuhn, Allan Lo

Year 2016
Pages 164
Publisher Amer Mathematical Society
Language en
ISBN 9781470420253
File Size 1.42 MB
File Format PDF
Download Counter 85
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Ebook Description

In this paper the authors prove the following results (via a unified approach) for all sufficiently large n: (i) [1-factorization conjecture] Suppose that n is even and D=>2 n/4 -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=> n/2 . 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=>n/2. Then G contains at least regeven (n,delta)/2=>(n-2)/8 edge-disjoint Hamilton cycles. Here regeven (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= n/2 of (iii) answer questions of Nash-Williams from 1970. All of the above bounds are best possible.