1-因子分解和Hamilton分解猜想的证明IV：两个团情况的例外系统
Proof of the 1-factorization and Hamilton decomposition conjectures IV: exceptional systems for the two cliques case
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。在目前的文件中，我们证明了稀疏图分解成路径系统的结果。当$G$接近于两个不相交的团的并集时，这些用于证明（i）和（ii）。
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 prove results on the decomposition of sparse graphs into path systems. These are used in the proof of (i) and (ii) in the case when $G$ is close to the union of two disjoint cliques.
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