Abstract
Given a graph G and \(X,Y\subset V(G)\), \(d_G(X,Y)\) is the distance between X and Y and the edge diameter \(diam_e(G)\) is the greatest distance between two edges of G. In this note, we consider edge diameter of a graph and its longest cycles and prove the following:
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(1)
Let G be a connected graph other than a tree with \(diam_e(G)\le d'\), then G has a longest cycle D such that \(d_G(e,D)\le d'-1\) for any edge e of G, furthermore, if G is 2-connected, then \(d_G(e,C)\le d'-1\) for any longest cycle C and any edge e of G.
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(2)
Let H be a 3-connected simple graph with \(diam_e(H)\ge d'\). Then H has a cycle of length at least \(2d'+3\) if H is not \(K_4\), furthermore, H has a cycle of length at least \(2d'+4\) if \(d'\ge 4\).
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Acknowledgements
The authors get this paper benefiting from the idea in manuscript [10] of Professor Akira Saito, we would like to thank him very much for sharing his work with us. Thank all referees for their comments which improve the presentation to the present version. This work is supported by Nature Science Funds of China (No. 12131013) and Beijing Natural Science Foundation (No. 1232005).
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Zhang, L., Xiong, L. & Tu, J. Edge-Diameter of a Graph and Its Longest Cycles. Graphs and Combinatorics 39, 89 (2023). https://doi.org/10.1007/s00373-023-02691-3
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DOI: https://doi.org/10.1007/s00373-023-02691-3