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A Note on Directed Genera of Some Tournaments

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Abstract

An embedding of a digraph in an orientable surface is an embedding as the underlying graph and arcs in each region force a directed cycle. The directed genus is the minimum genus of surfaces in which the digraph can be directed embedded. Bonnington, Conder, Morton and McKenna [J. Combin. Theory Ser. B, 85(2002) 1-20] gave the problem that which tournaments on n vertices have the directed genus ⌈(n−3)(n−4)/12 ⌉, the genus of Kn. In this paper, we use the current graph method to show that there exists a tournament, which has the directed genus ⌈(n−3)(n−4)/12 ⌉, on n vertices if and only if n ≡ 3 or 7 (mod 12).

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Authors and Affiliations

  1. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, China

    Jian-bing Liu & Rong-xia Hao

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  1. Jian-bing Liu
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  2. Rong-xia Hao
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Correspondence to Rong-xia Hao.

Additional information

Supported by the National Natural Science Foundation of China (No. 11731002) and the Fundamental Research Funds for the Central Universities (Nos. 2016JBM071, 2016JBZ012).

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Liu, Jb., Hao, Rx. A Note on Directed Genera of Some Tournaments. Acta Math. Appl. Sin. Engl. Ser. 34, 478–484 (2018). https://doi.org/10.1007/s10255-018-0763-9

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  • DOI: https://doi.org/10.1007/s10255-018-0763-9

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