Abstract
Let G be a simple graph of order \(n\ge 2\) and let \(k\in \{1,\ldots ,n-1\}\). The k-token graph \(F_k(G)\) of G is the graph whose vertices are the k-subsets of V(G), where two vertices are adjacent in \(F_k(G)\) whenever their symmetric difference is an edge of G. In 2018 Leaños and Trujillo-Negrete proved that if G is t-connected and \(t\ge k\), then \(F_k(G)\) is at least \(k(t-k+1)\)-connected. In this paper we show that such a lower bound remains true in the context of edge-connectivity. Specifically, we show that if G is t-edge-connected and \(t\ge k\), then \(F_k(G)\) is at least \(k(t-k+1)\)-edge-connected. We also provide some families of graphs attaining this bound.
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Acknowledgements
We thank A. L. Trujillo-Negrete for helpful discussions and by providing the example in Fig. 4. We also thank an anonymous referee for careful reading and improvements to the presentation.
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Leaños , J., Ndjatchi, C. The Edge-Connectivity of Token Graphs. Graphs and Combinatorics 37, 1013–1023 (2021). https://doi.org/10.1007/s00373-021-02301-0
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DOI: https://doi.org/10.1007/s00373-021-02301-0