Abstract A shortcut of a directed path v 1 v 2 ⋯ v n is an edge v i v j with j > i + 1 . If j = i + 2 the shortcut is a hop. If all hops are present, the path is called hop complete so the path and its hops form a square of a path. We prove that every tournament with n ≥ 4 vertices has a Hamiltonian path with at least ( 4 n − 10 ) ∕ 7 hops, and has a hop complete path of order at least n 0 . 295 . A spanning binary tree of a tournament is a spanning shortcut tree if for every vertex of the tree, all its left descendants are in-neighbors and all its right descendants are out-neighbors. It is well-known that every tournament contains a spanning shortcut tree. The number of shortcuts of a shortcut tree is the number of shortcuts of its unique induced Hamiltonian path. Let t ( n ) denote the largest integer such that every tournament with n vertices has a spanning shortcut tree with at least t ( n ) shortcuts. We almost determine the asymptotic growth of t ( n ) as it is proved that Θ ( n log 2 n ) ≥ t ( n ) − 1 2 n 2 ≥ Θ ( n log n ) .

中文翻译：

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锦标赛中有许多捷径

摘要 有向路径 v 1 v 2 ⋯ vn 的捷径是 j > i + 1 的边 vivj 。如果 j = i + 2，则快捷方式是一跳。如果所有跃点都存在，则该路径称为“跃点完成”，因此该路径及其跃点形成路径的正方形。我们证明每个具有 n ≥ 4 个顶点的锦标赛都有一个至少有 ( 4 n − 10 ) ∕ 7 跳的哈密顿路径，并且有一个阶数至少为 n 0 的跳完整路径。295 . 锦标赛的生成二叉树是生成快捷树，如果对于树的每个顶点，它的所有左后代都是in-neighbors，所有右后代是out-neighbors。众所周知，每个锦标赛都包含一个生成快捷树。捷径树的捷径数是其唯一的诱导哈密顿路径的捷径数。让 t ( n ) 表示最大的整数，使得每个具有 n 个顶点的锦标赛都有一个生成快捷方式树，其中至少有 t ( n ) 个快捷方式。我们几乎确定了 t ( n ) 的渐近增长，因为证明了 Θ (n log 2 n) ≥ t ( n ) − 1 2 n 2 ≥ Θ ( n log n ) 。