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On Universal Positive Graphs

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Abstract

We study the existence of the universal computable numberings and the universal graphs for various classes of positive graphs. It is known that each \( \forall \)-axiomatizable class of graphs \( K \) can be characterized as follows: A graph \( G \) belongs to \( K \) if and only if for a given family of finite graphs \( \mathbf{F} \) no graph in \( \mathbf{F} \) is isomorphically embeddable into \( G \).If all graphs in \( \mathbf{F} \) are weakly connected; then, under additional effectiveness conditions, the corresponding class \( K \) has some universal computable numbering and universal positive graph. The effectiveness conditions hold for forests, bipartite graphs, planar graphs, and \( n \)-colorable graphs (for a fixed number \( n \)). If \( \mathbf{F} \) is a finite family of the graphs with weakly connected complement then the corresponding class \( K \) contains a universal positive graph (in general, a universal computable numbering for \( K \) may fail to exist).

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Funding

The research was supported by the Science Committee of the Republic of Kazakhstan (Grant no. AP08856493 “Positive Graphs and Computable Reducibility on Them as Mathematical Models of Databases”). The work of Bazhenov was carried out in the framework of the State Task to the Sobolev Institute of Mathematics (Project FWNF–2022–0011).

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Authors and Affiliations

  1. Al-Farabi Kazakh National University, Almaty, Kazakhstan

    B. S. Kalmurzaev & D. B. Alish

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    N. A. Bazhenov

  3. Kazakh–British National University, Almaty, Kazakhstan

    B. S. Kalmurzaev

Authors
  1. B. S. Kalmurzaev
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  2. N. A. Bazhenov
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  3. D. B. Alish
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Corresponding author

Correspondence to B. S. Kalmurzaev.

Additional information

Translated from Sibirskii Matematicheskii Zhurnal, 2023, Vol. 64, No. 1, pp. 98–112. https://doi.org/10.33048/smzh.2023.64.110

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Kalmurzaev, B.S., Bazhenov, N.A. & Alish, D.B. On Universal Positive Graphs. Sib Math J 64, 83–93 (2023). https://doi.org/10.1134/S003744662301010X

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  • DOI: https://doi.org/10.1134/S003744662301010X

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