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Bipartite graphs and best proximity pairs

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Abstract

We say that a bipartite graph G(A, B) with the fixed parts A and B is proximinal if there is a semimetric space (X, d) such that A and B are disjoint proximinal subsets of X and all edges {a, b} satisfy the equality d(a, b) = dist(A, B). It is proved that a bipartite graph G is not isomorphic to any proximinal graph if and only if G is finite and empty. It is also shown that the subgraph induced by all non-isolated vertices of a nonempty bipartite graph G is a disjoint union of complete bipartite graphs if and only if G is isomorphic to a nonempty proximinal graph for an ultrametric space.

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Author information

Authors and Affiliations

  1. Ben M’sik Faculty of Sciences Hassan II University, Casablanca, Morocco

    Karim Chaira

  2. Institute of Applied Mathematics and Mechanics of NASU, Slovyansk, Ukraine

    Oleksiy Dovgoshey

  3. Institut fuer Mathematik Universitaet zu Luebeck, Luebeck, Deutschland

    Oleksiy Dovgoshey

  4. ENSAM Casablanca, Hassan II University, Casablanca, Morocco

    Samih Lazaiz

Authors
  1. Karim Chaira
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  2. Oleksiy Dovgoshey
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  3. Samih Lazaiz
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Corresponding author

Correspondence to Karim Chaira.

Additional information

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 19, No. 2, pp. 141–166, April–June, 2022.

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Chaira, K., Dovgoshey, O. & Lazaiz, S. Bipartite graphs and best proximity pairs. J Math Sci 264, 369–388 (2022). https://doi.org/10.1007/s10958-022-06005-5

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  • DOI: https://doi.org/10.1007/s10958-022-06005-5

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