Let f (n, k) denote the smallest number so that every connected graph with n vertices and minimum degree at least k contains a spanning tree in which the number of non-leaves is at most f (n, k). An early result of Linial and Sturtevant asserting that f (n, 3) = 3n/4 + O(1) and a related conjecture suggested by Linial led to a significant amount of work studying this function. It is known that for n much larger than k, \(f(n,k) \ge {n \over {k + 1}}(1 - \varepsilon (k))\ln (k + 1)\), where ε(k) tends to zero as k tends to infinity. Here we prove that \(f(n,k) \le {n \over {k + 1}}(\ln (k + 1) + 4) - 2\). This improves the error term in the best known upper bound for the function, due to Caro, West and Yuster, which is \(f(n,k) \le {n \over {k + 1}}(\ln (k + 1) + 0.5\sqrt {\ln (k + 1)} + 145)\). The proof provides an efficient deterministic algorithm for finding such a spanning tree in any given input graph satisfying the assumptions.

设f ( n, k ) 表示最小数，使得每个具有n 个顶点且最小度至少为k 的连通图都包含一棵生成树，其中非叶数最多为f ( n, k )。Linial 和 Sturtevant 断言f ( n , 3) = 3 n /4 + O (1) 的早期结果以及 Linial 提出的相关猜想引发了大量研究该函数的工作。已知对于n远大于k的情况，\(f(n,k) \ge {n \over {k + 1}}(1 - \varepsilon (k))\ln (k + 1)\)，其中ε当k趋于无穷大时， ( k ) 趋于零。这里我们证明\(f(n,k) \le {n \over {k + 1}}(\ln (k + 1) + 4) - 2\)。由于 Caro、West 和 Yuster，这改进了该函数最著名的上限中的误差项，即\(f(n,k) \le {n \over {k + 1}}(\ln (k + 1) + 0.5\sqrt {\ln (k + 1)} + 145)\)。该证明提供了一种有效的确定性算法，用于在任何给定的输入图中找到满足假设的生成树。