Abstract It was conjectured by Hoffmann–Ostenhof that the edge set of every connected cubic graph can be decomposed into a spanning tree, a matching and a family of disjoint cycles. In this paper, we show that the conjecture is true for connected claw-free cubic graphs, and, furthermore, any edge not contained any triangle appears only on the tree or on the matching. Then we show that the edge set of every connected cubic graph (except for K 4 and K 3 , 3 ) can be decomposed into a spanning tree and a family of disjoint paths of length at most 2.

中文翻译：

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连通无爪三次图的边分解

摘要 Hoffmann-Ostenhof 推测每个连通三次图的边集都可以分解为生成树、匹配和不相交环族。在本文中，我们证明了该猜想对于连通的无爪三次图是正确的，而且，任何不包含任何三角形的边只出现在树上或匹配上。然后我们证明每个连通三次图（除了 K 4 和 K 3 , 3 ）的边集都可以分解为一棵生成树和长度最多为 2 的不相交路径族。