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

Authors
  • Dobson, Edward1, 2
  • Hujdurović, Ademir2, 3
  • Kutnar, Klavdija2, 3
  • Morris, Joy4
  • 1 Mississippi State University, Department of Mathematics and Statistics, Mississippi State, MS, 39762, USA , Mississippi State (United States)
  • 2 University of Primorska, UP IAM, Muzejski trg 2, Koper, SI, 6000, Slovenia , Koper (Slovenia)
  • 3 University of Primorska, UP FAMNIT, Glagoljaška 8, Koper, SI, 6000, Slovenia , Koper (Slovenia)
  • 4 University of Lethbridge, Department of Mathematics and Computer Science, Lethbridge, AB, T1K 3M4, Canada , Lethbridge (Canada)
Type
Published Article
Journal
Journal of Algebraic Combinatorics
Publisher
Springer US
Publication Date
Sep 09, 2016
Volume
45
Issue
2
Pages
407–422
Identifiers
DOI: 10.1007/s10801-016-0711-9
Source
Springer Nature
Keywords
License
Yellow

Abstract

An automorphism α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} of a Cayley graph Cay(G,S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Cay}(G,S)$$\end{document} of a group G with connection set S is color-preserving if α(g,gs)=(h,hs)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha (g,gs) = (h,hs)$$\end{document} or (h,hs-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(h,hs^{-1})$$\end{document} for every edge (g,gs)∈E(Cay(G,S))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(g,gs)\in E(\mathrm{Cay}(G,S))$$\end{document}. If every color-preserving automorphism of Cay(G,S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Cay}(G,S)$$\end{document} is also affine, then Cay(G,S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Cay}(G,S)$$\end{document} is a Cayley color automorphism (CCA) graph. If every Cayley graph Cay(G,S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Cay}(G,S)$$\end{document} 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 F21\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{21}$$\end{document} of order 21. We first show that there is a unique non-CCA Cayley graph Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} of F21\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{21}$$\end{document}. We then show that if Cay(G,S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Cay}(G,S)$$\end{document} is a non-CCA graph of a group G of odd square-free order, then G=H×F21\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G = H\times F_{21}$$\end{document} for some CCA group H, and Cay(G,S)=Cay(H,T)□Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{Cay}(G,S) = \mathrm{Cay}(H,T)\mathbin {\square }\Gamma $$\end{document}.

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