Proof of the 1-Factorization and Hamilton Decomposition Conjectures (Memoirs of the American Mathematical Society) - Softcover

Csaba, Bela; Kuhn, Daniela; Lo, Allan; Osthus, Deryk; Treglown, Andrew

Softcover

ISBN 10: 1470420252 ISBN 13: 9781470420253

Publisher: American Mathematical Society, 2016

Synopsis

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, χ′(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 δ≥n/2. Then G contains at least regeven (n,δ)/2≥(n-2)/8 edge-disjoint Hamilton cycles. Here regeven (n,δ) denotes the degree of the largest even-regular spanning subgraph one can guarantee in a graph on n vertices with minimum degree δ.

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

About this title

About the Author

Bela Csaba, University of Szeged, Hungary.

Daniela Kuhn, University of Birmingham, United Kingdom.

Allan Lo, University of Birmingham, United Kingdom.

Deryk Osthus, University of Birmingham, United Kingdom.

Andrew Treglown, University of Birmingham, United Kingdom.

Edition details

• Publisher: American Mathematical Society
• Publication date: 2016
• Language: English
• ISBN 10: 1470420252
• ISBN 13: 9781470420253
• Binding: Paperback
• Number of pages: 164