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Spanning \(k\)-arc-strong subdigraphs with few arcs in \(k\)-arc-strong tournaments. (English) Zbl 1057.05039
It is shown that every \(k\)-arc-strong tournament \(T\) contains a spanning \(k\)-arc-strong subdigraph with at most \(nk+136k^2\) arcs. Moreover, this is best possible in terms of the exponent on \(k\). Such a subdigraph can be found in polynomial time. The proof uses new results on digraphs in which the number of non-neighbors of each vertex is bounded by some constant \(c\). It is conjectured that for every \(k\)-arc-strong tournament \(T\) the minimum number of arcs in a \(k\)-arc-strong spanning subdigraph of \(T\) is equal to the minimum number of arcs in a spanning subdigraph of \(T\) with the property that every vertex has in- and outdegree at least \(k\).

MSC:
05C20 Directed graphs (digraphs), tournaments
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