分裂有向图中的稀疏生成k-强定向子图
Sparse spanning $k$-strong oriented subdigraphs in split digraphs
AI总结：
本文针对分裂有向图类，证明了满足最小半度条件的k-强分裂有向图存在稀疏生成k-强定向子图，改进了k-强竞赛图的相关界，同时研究了Jackson与Thomassen的猜想及MSSS$_k$问题。
AI中文摘要：
Jackson与Thomassen猜想，每个2k-强有向图都存在一个生成k-强定向子图[《纽约科学院年报》555卷(1989)402-412页]。该猜想在k=1时成立，但除了针对有向图的某些特殊族（包括对称有向图）已获得的部分结果外，该猜想在一般情况下仍悬而未决。甚至连是否存在整数K，使得每个K-强有向图都有一个2-强生成定向子图这一问题也未解决。作为一个自然的优化对应问题，最小生成k-强子图（MSSS$_k$）问题是指，在一个k-强有向图中找到其生成k-强子图的最少弧数。该问题在k=1时已属于NP难问题，因为它推广了哈密顿圈问题。本文针对分裂有向图类同时研究上述两个问题，通过构造稀疏生成k-强定向子图来解决。具体而言，我们证明每个最小半度δ⁰(D)≥26k+15的k-强分裂有向图D=(V₁,V₂;A)，都包含一个生成k-强定向子图，其弧数不超过kn+k|V₁|+98k²+38k+3，其中kn+k|V₁|是紧的，k²项在常数因子范围内是紧的。对于最小半度至少为26k+15的k-强竞赛图类，我们的结果改进了Kang在《组合学、概率与计算》27卷(2018)892-907页中得到的界。
英文摘要：
Jackson and Thomassen conjectured that every $2k$-strong digraph admits a spanning $k$-strong oriented subdigraph [Ann. N. Y. Acad. Sci. 555 (1989) 402-412]. The conjecture holds for $k=1$ but other than some partial results that have been obtained for general $k$ in some special families of digraphs, including symmetric digraphs, the conjecture remains wide open in general. Even the existence of an integer $K$ such that every $K$-strong digraph has a 2-strong spanning oriented subdigraph is open. As a natural optimization counterpart, the minimum spanning $k$-strong subdigraph (MSSS$_k$) problem, is to find the minimum number of arcs in a spanning $k$-strong subdigraph of a $k$-strong digraph. This problem is NP-hard already for $k=1$ as it generalizes the hamiltonian cycle problem. In this paper, we address both problems simultaneously for the class of split digraphs, by constructing sparse spanning $k$-strong oriented subdigraphs. Specifically, we prove that every $k$-strong split digraph $D = (V_1, V_2; A)$ with minimum semi-degree {$δ^0(D)\geq 26k+15$} contains a spanning $k$-strong oriented subdigraph with no more than $kn+k|V_1|+98k^2+38k+3$ arcs, where $kn + k|V_1|$ is tight and the $k^{2}$ term is tight up to a constant factor. For the class of $k$-strong tournaments with minimum semi-degree at least $26k+15$ our results improve the bound obtained by Kang in [Combin. Probab. Comput., 27:892-907, 2018].
