1-因子分解和Hamilton分解结构III的证明：近似分解
Proof of the 1-factorization and Hamilton decomposition conjectures III: approximate decompositions
摘要：
在一系列的四篇论文中，我们证明了以下结果（通过一个统一的方法），所有足够大的$n$：（i）[1-因子分解猜想]假设$n$是偶数和$D\geq 2\lceil n/4\rceil -1$。则每一个在$n$顶点上的$D$ -正则图$G$都分解为完美匹配.等同物，$χ'(G)=D$。(ii)[Hamilton分解猜想]假设$D \ge \lfloor n/2 \rfloor $ .则每一个在$n$顶点上的$D$ -正则图$G$都有一个Hamilton圈的分解和至多一个完美匹配. (iii)本文证明了一个关于给定最小度的图中边不交Hamilton圈个数的最优结果。根据狄拉克的说法，（i）最早是在20世纪50年代提出的。(ii)（3）回答1970年Nash-Williams提出的问题。上述边界是最好的可能。本文证明了：假设$G$接近于完全平衡二部图或两个等长团的并.如果给我们一组合适的路系统，它覆盖了$G$的一组“例外”顶点和边，那么我们可以将这些路系统扩展为$G$到Hamilton圈（或完美匹配，如果合适的话）的近似分解。
英文摘要：
In a sequence of four papers, 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, $χ'(G)=D$. (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) We prove an optimal result on the number of edge-disjoint Hamilton cycles in a graph of given minimum degree. According to Dirac, (i) was first raised in the 1950s. (ii) and (iii) answer questions of Nash-Williams from 1970. The above bounds are best possible. In the current paper, we show the following: suppose that $G$ is close to a complete balanced bipartite graph or to the union of two cliques of equal size. If we are given a suitable set of path systems which cover a set of `exceptional' vertices and edges of $G$, then we can extend these path systems into an approximate decomposition of $G$ into Hamilton cycles (or perfect matchings if appropriate).
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