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arXiv · 2607.17116

K-Arc-Strong Orientations Of Semicomplete Digraphs

Tong Zhou·Jørgen Bang-Jensen·Jia Zhou·Jin Yan
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Abstract

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.

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Tong Zhou, Jørgen Bang-Jensen, Jia Zhou, Jin Yan. 2026-07-19. K-Arc-Strong Orientations Of Semicomplete Digraphs. https://arxiv.org/abs/2607.17116

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