Kelly, Luke; Kühn, Daniela; Osthus, Deryk A Dirac-type result on Hamilton cycles in oriented graphs. (English) Zbl 1172.05038 Comb. Probab. Comput. 17, No. 5, 689-709 (2008). 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. Cited in 11 Documents MSC: 05C45 Eulerian and Hamiltonian graphs 05C20 Directed graphs (digraphs), tournaments Citations:Zbl 0454.05029; Zbl 0795.05087 PDFBibTeX XMLCite \textit{L. Kelly} et al., Comb. Probab. Comput. 17, No. 5, 689--709 (2008; Zbl 1172.05038) Full Text: DOI arXiv References: [1] DOI: 10.2307/2308928 · Zbl 0089.39505 · doi:10.2307/2308928 [2] Thomassen, Surveys in Combinatorics 38 pp 211– (1979) [3] Komlós, Combinatorics: Paul Erd?s is Eighty pp 295– (1996) [4] DOI: 10.1007/BF01196135 · Zbl 0880.05049 · doi:10.1007/BF01196135 [5] Ghouila-Houri, CR Acad. Sci. Paris 25 pp 495– (1960) [6] DOI: 10.1017/S0963548398003502 · Zbl 0927.05041 · doi:10.1017/S0963548398003502 [7] DOI: 10.1016/j.jctb.2004.12.003 · Zbl 1061.05053 · doi:10.1016/j.jctb.2004.12.003 [8] DOI: 10.1007/s00493-003-0013-4 · Zbl 1046.05040 · doi:10.1007/s00493-003-0013-4 [9] DOI: 10.1017/S0963548307008395 · Zbl 1136.05029 · doi:10.1017/S0963548307008395 [10] Häggkvist, Combinatorics, Geometry and Probability pp 339– (1997) · doi:10.1017/CBO9780511662034.032 [11] DOI: 10.1016/0095-8956(90)90085-E · Zbl 0712.05030 · doi:10.1016/0095-8956(90)90085-E [12] Häggkvist, Combin. Probab. Comput. 2 pp 25– (1993) [13] Bollobás, Random Graphs (2001) · doi:10.1017/CBO9780511814068 [14] Bang-Jensen, Digraphs: Theory, Algorithms and Applications (2000) [15] Alon, The Probabilistic Method (2000) · doi:10.1002/0471722154 [16] DOI: 10.1016/j.jcss.2004.04.008 · Zbl 1084.68087 · doi:10.1016/j.jcss.2004.04.008 [17] DOI: 10.1112/plms/s3-24.4.739 · Zbl 0233.05117 · doi:10.1112/plms/s3-24.4.739 [18] DOI: 10.1112/plms/s3-45.1.151 · Zbl 0486.05049 · doi:10.1112/plms/s3-45.1.151 [19] DOI: 10.1112/plms/s3-42.2.231 · Zbl 0454.05029 · doi:10.1112/plms/s3-42.2.231 [20] DOI: 10.1007/BF02128671 · Zbl 0729.05021 · doi:10.1007/BF02128671 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.