Abstract
An edge-transitive graph \(\varGamma \) is called alternatively connected if a subgroup G of the automorphism group of \(\varGamma \) has two orbits on the arc set of \(\varGamma \), and there exists an alternative walk (with respect to a given G-orbit on arcs) between every pair of vertices of \(\varGamma \). Employing the standard double covers of digraphs, we give some basic properties of alternatively connected edge-transitive graphs. The main result of this paper is a reduction result on alternatively connected edge-transitive graphs of square-free order. As an application of this result, we give a characterization for alternatively connected edge-transitive graphs of square-free order and valency 6. It is proved that such a graph is either a circulant or constructed from \(\mathrm{PSL}(2,p)\).
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Supported by National Natural Science Foundation of China (11371204, 11731002).
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Li, G., Lu, Z.P. On Alternatively Connected Edge-Transitive Graphs of Square-free Order. Graphs and Combinatorics 33, 1577–1589 (2017). https://doi.org/10.1007/s00373-017-1855-7
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DOI: https://doi.org/10.1007/s00373-017-1855-7