AI 中文总结
研究半完全有向图推广中高度连通生成有向子图问题，证明扩展半完全有向图中$(4k + 1)$-强时有生成$k$-强有向子图，半完全分裂有向图中$5k$-强时有该子图，得出相关$f(k)$存在性结论及对一类有向图的$f(k)$上限。
AI 中文摘要
1989年，Jackson和Thomassen猜想存在整数函数$f(k)$，使得每个$f(k)$-强有向图都有一个生成$k$-强有向子图，甚至猜想$f(k)=2k$。对于一般有向图，$f(2)$的存在性仍是开放问题，Thomassen证明对称有向图中$f(2)=4$。对于一般的$k$，仅知道局部半完全有向图和拟传递有向图中$f(k)$的存在性。Guo证明每个$(3k - 2)$-强局部半完全有向图包含一个生成$k$-强局部竞赛图。本文证明了半完全多部有向图的两个子类，即扩展半完全有向图和半完全分裂有向图中$f(k)$的存在性。证明了每个$(4k + 1)$-强扩展半完全有向图包含一个生成$k$-强有向子图，每个$5k$-强半完全分裂有向图包含一个生成$k$-强有向子图。第一个结果意味着对于一大类可通过用任意有向图替换至少3个顶点的半完全有向图$S$的每个顶点得到的有向图，也有$f(k)\leq 4k + 1$。
英文摘要
Let $k$ be a positive integer. Jackson and Thomassen conjectured in 1989 that there exists an integer function $f(k)$ such that every $f(k)$-strong digraph admits a spanning $k$-strong oriented subdigraph. They even conjectured that one can take $f(k)=2k$ [Ann. N. Y. Acad. Sci. 555 (1989) 402-412]. Already the existence of $f(2)$ is open for general digraphs. Thomassen proved that $f(2)=4$ for symmetric digraphs. For general $k$, the existence of $f(k)$ was only known for locally semicomplete digraphs and quasi-transitive digraphs. Guo proved that every ${(3k-2)}$-strong locally semicomplete digraph contains a spanning $k$-strong local tournament [Discrete Appl. Math. 79 (1997) 119--125]. One can deduce from Guo's result that we have $f(k)\leq 3k-2$ for quasi-transitive digraphs. In this paper, we prove the existence of $f(k)$ for two subclasses of the semicomplete multipartite digraphs, namely extended semicomplete digraphs and semicomplete split digraphs. We prove that every $(4k+1)$-strong extended semicomplete digraph contains a spanning $k$-strong oriented subdigraph and every $5k$-strong semicomplete split digraph contains a spanning $k$-strong oriented subdigraph. The first result implies that for the large class of digraphs which can be obtained from some semicomplete digraph $S$ on at least 3 vertices by substituting arbitrary digraphs for each vertex of $S$ we also have $f(k)\leq 4k+1$.
Comments17pages, 1 figure
