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The maximum number of induced open triangles in graphs of a given order

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Abstract

An open triangle is a simple, undirected graph consisting of three vertices and two edges. It is shown that the maximum number of induced open triangles in a graph on n vertices is given by \(\left( \frac{n}{2}-1\right) \lfloor \frac{n}{2}\rfloor \lceil \frac{n}{2}\rceil \). The maximum is achieved for the complete bipartite graph \(K_{\lfloor {n}/{2}\rfloor , \lceil {n}/{2}\rceil }\). The maximum expected number of open triangles in a uniform random graph on n vertices is observed to be asymptotically equivalent.

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Acknowledgements

The authors thank referees for their helpful remarks. The work of the first author was partially supported by the project 17-01-00170 of Russian Foundation for Basic Research and by the program of fundamental scientific researches of the SB RAS I.5.1., Project 0314-2016-0014. Partial support of the second and third authors by NSF Grant CMMI-1538493 is also gratefully acknowledged.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, 4 Koptyug Ave., Novosibirsk, Russia, 630090

    Artem Pyatkin

  2. Novosibirsk State University, 2 Pirogova Str., Novosibirsk, Russia, 630090

    Artem Pyatkin

  3. Department of Industrial and Systems Engineering, Texas A&M University, College Station, TX, 77843, USA

    Eugene Lykhovyd & Sergiy Butenko

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  1. Artem Pyatkin
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  2. Eugene Lykhovyd
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  3. Sergiy Butenko
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Correspondence to Sergiy Butenko.

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Pyatkin, A., Lykhovyd, E. & Butenko, S. The maximum number of induced open triangles in graphs of a given order. Optim Lett 13, 1927–1935 (2019). https://doi.org/10.1007/s11590-018-1330-2

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