Abstract
We introduce the concept of the asymmetry number for finite digraphs, as a natural generalization of that for undirected graphs by Erdős and Rényi in 1963. We prove an upper bound for the asymmetry number of finite digraphs and give a condition for equality. We show that our bound is asymptotically best for digraphs with sufficiently large order. We also consider the random oriented graph , and make some remarks on Aut.
Funding Statement
This research is supported by Grant-in-Aid for JSPS Fellows 18J11282, Grant-in-Aid for Young Scientists (B) 26870259 and Grant-in-Aid for Scientific Research (B) 15H03636 of the Japan Society for the Promotion of Science.
Acknowledgement
We would appreciate Hikoe Enomoto and Masatake Hirao for valuable comments and suggestions. The authors would also like to thank Peter J. Cameron for his careful reading of our paper.
Citation
Shohei Satake. Masanori Sawa. Masakazu Jimbo. "Erdős-Rényi theory for asymmetric digraphs." SUT J. Math. 54 (2) 109 - 129, December 2018. https://doi.org/10.55937/sut/1547570388
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