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Out-arc pancyclicity of vertices in tournaments. (English) Zbl 1220.05049
Summary: T. Yao, Y. Guo and K. Zhang [“Pancyclic out-arcs of a vertex in a tournament,” Discrete Appl. Math. 99, No. 1-3, 245–249 (2000; Zbl 0939.05045)] proved that every strong tournament contains a vertex \(u\) such that every out-arc of \(u\) is pancyclic. In this paper, we prove that every strong tournament with minimum out-degree at least two contains two such vertices. A. Yeo [“The number of pancyclic arcs in a \(k\)-strong tournament,” J. Graph Theory 50, No. 3, 212–219 (2005; Zbl 1081.05041)] conjectured that every 2-strong tournament has three distinct vertices \(\{ x,y,z \}\), such that every arc out of \(x,y\) and \(z\) is pancyclic. In this paper, we also prove that Yeo’s conjecture is true.

05C20 Directed graphs (digraphs), tournaments
05C38 Paths and cycles
Full Text: DOI
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