Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$ , improving a result of Han, who showed that in this range almost all vertices of G can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if $d = \Omega(n)$ and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite graphs).

中文翻译：

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正则图的循环分区

Magnant 和 Martin 猜想任何d- 正则图G在n顶点可以划分为$n / (d+1)$路径（存在一个简单的结构，表明这个界限是最好的）。我们证明这个猜想时$d = \Omega(n)$，改进了 Han 的结果，他表明在这个范围内几乎所有的顶点G可以覆盖$n / (d+1) + 1$顶点不相交的路径。事实上，我们的证明给出了一个划分五(G) 成循环。我们还表明，如果$d = \Omega(n)$和G是二分的，那么五(G) 可以划分为n/(2d) 路径（这个界限对于二分图来说很紧）。