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Intersecting Longest Cycles in Archimedean Tilings

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Abstract

In 1966 Gallai asked the following question: Do all longest paths (cycles) of a connected graph contain a common vertex? After a positive answer to Gallai’s question, another question has been raised, Is there any family of graphs without Gallai’s property? Menke found one such family, the square lattices. Embedding methods hold the promise to transform not just the way calculations are performed, but to significantly reduce computational costs and often used in quantum mechanics and material sciences. In this paper, we prove the existence of graphs with the empty intersection of their longest cycles as subgraphs of Archimedean lattices.

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Authors and Affiliations

  1. Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore, Pakistan

    Muhammad Faisal Nadeem, Hamza Iqbal & Hafiz Muhammad Afzal Siddiqui

  2. Department of Mathematics, Riphah Institute of Computing and Applied Sciences, Riphah International University, Lahore, Pakistan

    Muhammad Azeem

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  1. Muhammad Faisal Nadeem
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  2. Hamza Iqbal
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  3. Hafiz Muhammad Afzal Siddiqui
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  4. Muhammad Azeem
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Correspondence to Muhammad Faisal Nadeem.

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Nadeem, M.F., Iqbal, H., Siddiqui, H.M.A. et al. Intersecting Longest Cycles in Archimedean Tilings. Algorithmica 85, 2348–2362 (2023). https://doi.org/10.1007/s00453-023-01104-4

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