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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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##### References:
 [1] Bang-Jensen, J.; Gutin, G., Digraphs: theory, algorithms and applications, (2001), Springer-Verlag London Limited · Zbl 0958.05002 [2] Bermond, J.C.; Thomassen, C., Cycles in digraphs—a survey, J. graph theory, 5, 1-43, (1981) · Zbl 0458.05035 [3] Caccetta, L.; Häggkvist, R., On minimal digraphs with given girth, (), 181-187 · Zbl 0406.05033 [4] Hamidoune, Y.O., A note on minimal directed graphs with given girth, J. combin. theory ser. B, 43, 343-348, (1987) · Zbl 0643.05036 [5] Havet, F.; Thomassé, S.; Yeo, A., Hoàng – reed conjecture holds for tournaments, Discrete math., 308, 3412-3415, (2008) · Zbl 1147.05038 [6] Hoàng, C.T.; Reed, B., A note on short cycles in digraphs, Discrete math., 66, 103-107, (1987) · Zbl 0626.05021 [7] B.D. Sullivan, A summary of results and problems related to the Caccetta-Häggkvist conjecture, preprint. [8] Thomassen, C., Disjoint cycles in digraphs, Combinatorica, 3, 393-396, (1983) · Zbl 0527.05036 [9] Thomassen, C., The 2-linkage problem for acyclic digraphs, Discrete math., 55, 73-87, (1985) · Zbl 0563.05027 [10] Thomassen, C., Even cycles in directed graphs, European J. combin., 6, 85-89, (1985) · Zbl 0606.05039
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