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Squaring the tournament—an open problem. (English) Zbl 0904.05034
Summary: Let $$T$$ be a tournament. We define the digraph $$T^2$$ on the vertices of $$T$$ as follows: For all vertices $$x$$ and $$y$$ of $$T$$, if $$(x,y)$$ is an arc in $$T$$ then $$(x,y)$$ is an arc in $$T^2$$ and if $$(x,y)$$ is an arc in $$T$$ and there is a vertex $$z$$ making $$\{x,y,z\}$$ a 3-cycle in $$T$$ then $$(y,x)$$ is an arc in $$T^2$$.
Dean has conjectured that there is some vertex whose outdegree in $$T$$ is at least doubled in $$T^2$$. This is a special case of a conjecture of Seymour’s for oriented graphs. We prove Dean’s conjecture for several families of tournaments.

##### MSC:
 05C20 Directed graphs (digraphs), tournaments
##### Keywords:
tournament; digraph