A characterisation is given of edge-transitive Cayley graphs of valency 4 on odd number of vertices. The characterisation is then applied to solve several problems in the area of edge-transitive graphs: answering a question proposed by Xu [Automorphism groups and isomorphisms of Cayley graphs, Discrete Math. 182 (1998) 309-319] regarding normal Cayley graphs; providing a method for constructing edge-transitive graphs of valency 4 with arbitrarily large vertex-stabiliser; constructing and characterising a new family of half-transitive graphs. Also this study leads to a construction of the first family of arc-transitive graphs of valency 4 which are non-Cayley graphs and have a 'nice' isomorphic 2-factorisation. (c) 2005 Elsevier Inc. All rights reserved.
Li, C-H., Lu, Z. P., & Zhang, H. (2006). Tetravalent edge-transitive Cayley graphs with odd number of vertices. Journal of combinatorial Theory Series B, 96(1), 164-181. https://doi.org/10.1016/j.jctb.2005.07.003