On color-preserving automorphisms of Cayley graphs of odd square-free order

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• 作者：Edward Dobson ; Ademir Hujdurović ; Klavdija Kutnar…
• 关键词：Cayley graph ; Color ; preserving automorphism ; Automorphism group ; Affine automorphism
• 刊名：Journal of Algebraic Combinatorics
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：45
• 期：2
• 页码：407-422
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Combinatorics; Convex and Discrete Geometry; Order, Lattices, Ordered Algebraic Structures; Computer Science, general; Group Theory and Generalizations;
• 出版者：Springer US
• ISSN：1572-9192
• 卷排序：45

文摘

An automorphism \(\alpha \) of a Cayley graph \(\mathrm{Cay}(G,S)\) of a group G with connection set S is color-preserving if \(\alpha (g,gs) = (h,hs)\) or \((h,hs^{-1})\) for every edge \((g,gs)\in E(\mathrm{Cay}(G,S))\). If every color-preserving automorphism of \(\mathrm{Cay}(G,S)\) is also affine, then \(\mathrm{Cay}(G,S)\) is a Cayley color automorphism (CCA) graph. If every Cayley graph \(\mathrm{Cay}(G,S)\) is a CCA graph, then G is a CCA group. Hujdurović et al. have shown that every non-CCA group G contains a section isomorphic to the non-abelian group \(F_{21}\) of order 21. We first show that there is a unique non-CCA Cayley graph \(\Gamma \) of \(F_{21}\). We then show that if \(\mathrm{Cay}(G,S)\) is a non-CCA graph of a group G of odd square-free order, then \(G = H\times F_{21}\) for some CCA group H, and \(\mathrm{Cay}(G,S) = \mathrm{Cay}(H,T)\mathbin {\square }\Gamma \).