In this paper, we propose a new conjecture that the complete graph \(K_{4m+1}\) can be decomposed into copies of two arbitrary trees, each of size \(m, m \ge 1\). To support this conjecture we prove that the complete graph \(K_{4cm+1}\) can be decomposed into copies of an arbitrary tree with m edges and copies of the graph H, where H is either a path with m edges or a star with m edges and where c is any positive integer. Further, we discuss related open problems.

中文翻译：

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将完整图分解为任意树

在本文中，我们提出了一个新的猜想，可以将完整图\（K_ {4m + 1} \）分解为两个任意树的副本，每个树的大小为\（m，m \ ge 1 \）。为了支持这个猜想，我们证明了完整的图\（K_ {4cm + 1} \）可以分解为具有m个边的任意树的副本和图H的副本，其中H是具有m个边的路径或a带有m条边的星形，其中c是任何正整数。此外，我们讨论了相关的公开问题。