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The Hoàng-Reed conjecture for \(\delta ^{+}=3\). (English) Zbl 1222.05091
Summary: The Hoàng-Reed Conjecture states that a digraph with minimum out-degree \(d\) contains \(d\) dicycles \(C_{1},C_{2},\dots ,C_d\) such that \(C_k\) intersects \(\bigcup _{i<k}C_i\) on at most one vertex, for each \(k\). It was made as a more structural approach to the Caccetta-Häggkvist Conjecture. We introduce the concept of a nearest separator and use it to prove a stronger version of the Hoàng-Reed Conjecture for the \(\delta ^{+}=2\) case. With these two tools we go on to prove the \(\delta ^{+}=3\) case.

MSC:
05C20 Directed graphs (digraphs), tournaments
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