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On \(\ell \)-Distance-Balanced Product Graphs

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Abstract

A graph G is \(\ell \)-distance-balanced if for each pair of vertices x and y at a distance \(\ell \) in G, the number of vertices closer to x than to y is equal to the number of vertices closer to y than to x. A complete characterization of \(\ell \)-distance-balanced corona products is given and characterization of lexicographic products for \(\ell \ge 3\), thus complementing known results for \(\ell \in \{1,2\}\) and correcting an earlier related assertion. A sufficient condition on H which guarantees that \(K_n\,\square \,H\) is \(\ell \)-distance-balanced is given, and it is proved that if \(K_n\,\square \,H\) is \(\ell \)-distance-balanced, then H is an \(\ell \)-distance-balanced graph. A known characterization of 1-distance-balanced graphs is extended to \(\ell \)-distance-balanced graphs, again correcting an earlier claimed assertion.

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Acknowledgements

We thank referees for numerous useful remarks which, in particular, enabled us to significantly shorten some of the arguments. We acknowledge the financial support from the Slovenian Research Agency (research core funding no. P1-0297 and projects J1-8130, J1-9109, J1-1693, N1-0095).

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Authors and Affiliations

  1. Faculty of Organizational Sciences, University of Maribor, Maribor, Slovenia

    Janja Jerebic & Gregor Rus

  2. Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor, Slovenia

    Janja Jerebic & Sandi Klavžar

  3. Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

    Sandi Klavžar

  4. Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

    Sandi Klavžar & Gregor Rus

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  1. Janja Jerebic
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  2. Sandi Klavžar
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Correspondence to Sandi Klavžar.

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Jerebic, J., Klavžar, S. & Rus, G. On \(\ell \)-Distance-Balanced Product Graphs. Graphs and Combinatorics 37, 369–379 (2021). https://doi.org/10.1007/s00373-020-02247-9

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  • DOI: https://doi.org/10.1007/s00373-020-02247-9

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