Abstract
In an earlier paper (see Sali and Simonyi Eur. J. Combin. 20, 93–99, 1999) the first two authors have shown that self-complementary graphs can always be oriented in such a way that the union of the oriented version and its isomorphically oriented complement gives a transitive tournament. We investigate the possibilities of generalizing this theorem to decompositions of the complete graph into three or more isomorphic graphs. We find that a complete characterization of when an orientation with similar properties is possible seems elusive. Nevertheless, we give sufficient conditions that generalize the earlier theorem and also imply that decompositions of odd vertex complete graphs to Hamiltonian cycles admit such an orientation. These conditions are further generalized and some necessary conditions are given as well.
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The first author’s work is partially supported by the National Research, Development and Innovation Office (NKFIH) grant K–116769 and K–132696. The second author’s work is partially supported by the National Research, Development and Innovation Office (NKFIH) grants K–116769, K–120706 and K–132696. The third author’s work is partially supported by the Cryptography “Lendület” project of the Hungarian Academy of Sciences and by the National Research, Development and Innovation Office (NKFIH) grants K–116769, K–132696 and SSN-117879, ERC Synergy grant “Dynasnet”, and the grant of the Russian Government N075-15-2019-1926. The first two authors’ work is also supported by the National Research, Development and Innovation Fund (TUDFO/51757/2019-ITM, Thematic Excellence Program), the BME NC TKP2020 grant of NKFIH Hungary and it is also connected to the scientific program of the “Development of quality-oriented and harmonized R+D+I strategy and functional model at BME” project, supported by the New Hungary Development Plan (Project ID: TÁMOP-4.2.1/B-09/1/KMR-2010-0002).
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Sali, A., Simonyi, G. & Tardos, G. Partitioning Transitive Tournaments into Isomorphic Digraphs. Order 38, 211–228 (2021). https://doi.org/10.1007/s11083-020-09535-2
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DOI: https://doi.org/10.1007/s11083-020-09535-2