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A Dirac-type result on Hamilton cycles in oriented graphs. (English) Zbl 1172.05038

Summary: We show that for each \(\alpha>0\) every sufficiently large oriented graph \(G\) with \(\delta^+(G), \delta^-(G)\geq 3|G|/8+ \alpha|G|\) contains a Hamilton cycle. This gives an approximate solution to a problem of C. Thomassen [Proc. Lond. Math. Soc., III. Ser. 42, 231–251 (1981; Zbl 0454.05029)]. In fact, we prove the stronger result that \(G\) is still Hamiltonian if \(\delta(G)+ \delta^+(G)+ \delta^-(G)\geq 3|G|/2+ \alpha|G|\). Up to the term \(\alpha|G|\), this confirms a conjecture of R. Häggkvist [Comb. Probab. Comput. 2, No. 1, 25–32 (1993; Zbl 0795.05087)]. We also prove an Ore-type theorem for oriented graphs.

MSC:

05C45 Eulerian and Hamiltonian graphs
05C20 Directed graphs (digraphs), tournaments
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