Semiregular automorphisms in vertex-transitive graphs with a solvable group of automorphisms. (English) Zbl 1381.20005
Summary: It has been conjectured that automorphism groups of vertex-transitive (di)graphs, and more generally 2-closures of transitive permutation groups, must necessarily possess a fixed-point-free element of prime order, and thus a non-identity element with all orbits of the same length, in other words, a semiregular element. The known affirmative answers for graphs with primitive and quasiprimitive groups of automorphisms suggest that solvable groups need to be considered if one is to hope for a complete solution of this conjecture. It is the purpose of this paper to present an overview of known results and suggest possible further lines of research towards a complete solution of the problem.
MSC:
| 20B25 | Finite automorphism groups of algebraic, geometric, or combinatorial structures |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
Keywords:
solvable group; semiregular automorphism; fixed-point-free automorphism; polycirculant conjectureReferences:
| [1] | B. Alspach, Lifting Hamilton cycles of quotient graphs, Discrete Math. 78 (1989), 25-36, doi: 10.1016/0012-365x(89)90157-x. · Zbl 0702.05054 |
| [2] | P. J. Cameron, M. Giudici, G. A. Jones, W. M. Kantor, M. H. Klin, D. Maruˇsiˇc and L. A. Nowitz, Transitive permutation groups without semiregular subgroups, J. London Math. Soc. 66 (2002), 325-333, doi:10.1112/s0024610702003484. · Zbl 1015.20001 |
| [3] | P. J. Cameron, J. Sheehan and P. Spiga, Semiregular automorphisms of vertex-transitive cubic graphs, European J. Combin. 27 (2006), 924-930, doi:10.1016/j.ejc.2005.04.008. · Zbl 1090.05032 |
| [4] | P. J. Cameron (ed.), Problems from the Fifteenth British Combinatorial Conference, Discrete Math.167/168 (1997), 605-615, doi:10.1016/s0012-365x(96)00212-9. · Zbl 0870.05075 |
| [5] | E. Dobson, A. Malniˇc, D. Maruˇsiˇc and L. A. Nowitz, Minimal normal subgroups of transitive permutation groups of square-free degree, Discrete Math. 307 (2007), 373-385, doi:10.1016/j. disc.2005.09.029. · Zbl 1110.05043 |
| [6] | E. Dobson, A. Malniˇc, D. Maruˇsiˇc and L. A. Nowitz, Semiregular automorphisms of vertextransitive graphs of certain valencies, J. Combin. Theory Ser. B 97 (2007), 371-380, doi:10. 1016/j.jctb.2006.06.004. · Zbl 1113.05046 |
| [7] | E. Dobson and D. Maruˇsiˇc, On semiregular elements of solvable groups, Comm. Algebra 39 (2011), 1413-1426, doi:10.1080/00927871003738923. · Zbl 1232.20004 |
| [8] | B. Fein, W. M. Kantor and M. Schacher, Relative Brauer groups II, J. Reine Angew. Math. 328 (1981), 39-57. · Zbl 0457.13004 |
| [9] | M. Giudici, Quasiprimitive groups with no fixed point free elements of prime order, J. London Math. Soc.67 (2003), 73-84, doi:10.1112/s0024610702003812. · Zbl 1050.20002 |
| [10] | M. Giudici, New constructions of groups without semiregular subgroups, Comm. Algebra 35 (2007), 2719-2730, doi:10.1080/00927870701353357. · Zbl 1155.20001 |
| [11] | M. Giudici, P. Potoˇcnik and G. Verret, Semiregular automorphisms of edge-transitive graphs, J. Algebraic Combin.40 (2014), 961-972, doi:10.1007/s10801-014-0515-8. · Zbl 1304.05059 |
| [12] | M. Giudici and J. Xu, All vertex-transitive locally-quasiprimitive graphs have a semiregular automorphism, J. Algebraic Combin. 25 (2007), 217-232, doi:10.1007/s10801-006-0032-5. · Zbl 1112.05050 |
| [13] | A. Hujdurovi´c, K. Kutnar and D. Maruˇsiˇc, Odd automorphisms in vertex-transitive graphs, Ars Math. Contemp.10 (2016), 427-437,http://amc-journal.eu/index.php/amc/ article/view/1046. · Zbl 1351.05106 |
| [14] | K. Kutnar and D. Maruˇsiˇc, Recent trends and future directions in vertex-transitive graphs, Ars Math. Contemp.1 (2008), 112-125,http://amc-journal.eu/index.php/amc/ article/view/81. · Zbl 1163.05025 |
| [15] | K. Kutnar and D. Maruˇsiˇc, A complete classification of cubic symmetric graphs of girth 6, J. Combin. Theory Ser. B99 (2009), 162-184, doi:10.1016/j.jctb.2008.06.001. · Zbl 1205.05193 |
| [16] | K. Kutnar, D. Maruˇsiˇc, P. ˇSparl, R.-J. Wang and M.-Y. Xu, Classification of half-arc-transitive graphs of order 4p, European J. Combin. 34 (2013), 1158-1176, doi:10.1016/j.ejc.2013.04. 004. · Zbl 1292.05134 |
| [17] | L. Lov´asz, Problem 11, in: R. Guy, H. Hanani, N. Sauer and J. Schonheim (eds.), Combinatorial Structures and Their Applications, Gordon & Breach, New York, 1970 pp. 243-246, proceedings of the Calgary International Conference on Combinatorial Structures and their Applications held at the University of Calgary (Calgary, Alberta, Canada, 1969). · Zbl 0257.05122 |
| [18] | D. Maruˇsiˇc, On vertex symmetric digraphs, Discrete Math. 36 (1981), 69-81, doi:10.1016/ 0012-365x(81)90174-6. |
| [19] | D. Maruˇsiˇc and R. Scapellato, Permutation groups, vertex-transitive digraphs and semiregular automorphisms, European J. Combin. 19 (1998), 707-712, doi:10.1006/eujc.1997.0192. · Zbl 0912.05038 |
| [20] | A. Ramos Rivera and P. ˇSparl, The classification of half-arc-transitive generalizations of Bouwer graphs, European J. Combin. 64 (2017), 88-112, doi:10.1016/j.ejc.2017.04.003. · Zbl 1365.05129 |
| [21] | P. Spiga, Semiregular elements in cubic vertex-transitive graphs and the restricted Burnside problem, Math. Proc. Cambridge Philos. Soc. 157 (2014), 45-61, doi:10.1017/ s0305004114000188. · Zbl 1297.05113 |
| [22] | P. ˇSparl, A classification of tightly attached half-arc-transitive graphs of valency 4, J. Combin. Theory Ser. B98 (2008), 1076-1108, doi:10.1016/j.jctb.2008.01.001. · Zbl 1158.05030 |
| [23] | G. Verret, Arc-transitive graphs of valency 8 have a semiregular automorphism, Ars Math. Contemp.8 (2015), 29-34,http://amc-journal.eu/index.php/amc/article/ view/492. · Zbl 1318.05090 |
| [24] | J. Xu, Semiregular automorphisms of arc-transitive graphs with valency pq, European J. Combin.29 (2008), 622-629, doi:10.1016/j.ejc.2007.04.008. · Zbl 1142.05044 |
| [25] | S. Zhou, Classification of a family of symmetric graphs with complete 2-arc-transitive quotients, Discrete Math. 309 (2009), 5404-5410, doi:10.1016/j.disc.2008.12.001. · Zbl 1198.05098 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
