Proof of the 1-factorization and Hamilton decomposition conjectures
著者
書誌事項
Proof of the 1-factorization and Hamilton decomposition conjectures
（Memoirs of the American Mathematical Society, no. 1154）
American Mathematical Society, 2016
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注記
"Volume 244, number 1154 (third of 4 numbers), November 2016"
Includes bibliographical references
内容説明・目次
内容説明
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.
目次
Introduction
The two cliques case
Exceptional systems for the two cliques case
The bipartite case
Approximate decompositions
Bibliography
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