半完全有向图的K-弧强定向
K-Arc-Strong Orientations Of Semicomplete Digraphs
AI总结：
研究半完全有向图的K-弧强定向问题，利用弗兰克的一般定向定理及基于半完全结构的计数论证，证明至少2k+1个顶点的(2k-1)-弧强半完全有向图含生成的k-弧强锦标赛，给出2k-1和2k+1是最佳界限。
AI中文摘要：
杰克逊和弗兰克的结果表明，每个2k-弧强有向图D都包含一个生成的k-弧强定向子图，即使对于非常密集的有向图，这也是最好的情况。如果对于每对不同的顶点x,y，至少存在弧xy或yx之一，则有向图是半完全的。锦标赛对于每对这样的顶点恰好有xy或yx之一。显然，每个半完全有向图D都包含一个生成锦标赛T，它是通过从D的每个2-循环中删除一条弧得到的。我们证明，每个至少有2k+1个顶点的(2k-1)-弧强半完全有向图都包含一个生成的k-弧强锦标赛。2k-1和2k+1这两个界限都是最好的情况。证明使用了弗兰克关于图的一般定向定理以及基于半完全结构的计数论证。
英文摘要：
Results by Jackson and Frank imply that every 2k-arc-strong digraph D contains a spanning k-arc-strong oriented subdigraph. This is best possible, even for very dense digraphs. A digraph is semicomplete if at least one of the arcs xy,yx is present for every pair of distinct vertices x,y. A tournament has exactly one of xy,yx for every such pair. Clearly every semicomplete digraph D contains a spanning tournament T which is obtained by deleting one arc from every 2-cycle of D. We prove that every (2k-1)-arc-strong semicomplete digraph on at least 2k+1 vertices contains a spanning k-arc-strong tournament. Both bounds 2k-1 and 2k+1 are best possible. The proof uses Frank's general orientation theorem for graphs as well as counting arguments based on the semicomplete structure.
