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首页> 外文期刊>Journal of Combinatorial Theory, Series B >Symmetric cubic graphs of small girth
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Symmetric cubic graphs of small girth

机译：小周长的对称立方图

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A graph F is symmetric if its automorphism group acts transitively on the arcs of F, and s-regular if its automorphism group acts regularly on the set of s-arcs of F. Tutte [WT. Tutte, A family of cubical graphs, Proc. Cambridge Philos. Soc. 43 (1947) 459-474; W.T. Tutte, On the symmetry of cubic graphs, Canad. J. Math. 11 (1959) 621-624] showed that every cubic finite symmetric cubic graph is s-regular for some s <= 5. We show that a symmetric cubic graph of girth at most 9 is either I-regular or 2 '-regular (following the notation of Djokovic), or belongs to a small family of exceptional graphs. On the other hand, we show that there are infinitely many 3-regular cubic graphs of girth 10, so that the statement for girth at most 9 cannot be improved to cubic graphs of larger girth. Also we give a characterisation of the 1- or 2 '-regular cubic graphs of girth g <= 9, proving that with five exceptions these are closely related with quotients of the triangle group Delta (2, 3, g) in each case, or of the group (x, y vertical bar x(2) = y(3) = [x, y](4) = 1) in the case g = 8. All the 3-transitive cubic graphs and exceptional 1- and 2-regular cubic graphs of girth at most 9 appear in the list of cubic symmetric graphs up to 768 vertices produced by Conder and Dobcsanyi [M. Conder, P. Dobcsanyi, Trivalent symmetric graphs up to 768 vertices, J. Combin. Math. Combin. Comput. 40 (2002) 41-63]; the largest is the 3-regular graph F570 of order 570 (and girth 9). The proofs of the main results are computer-assisted. (c) 2007 Elsevier Inc. All rights reserved.
机译：如果图F的自同构群在F的弧上传递，则图是对称的;如果图的自同构群在F. s弧的集合上规则地作用，则s-regular。 Tutte，立方图族，Proc。剑桥Philos。 Soc。 43（1947）459-474;图蒂（W.T. Tutte），关于三次图的对称性，加拿大。 J.数学11（1959）621-624]表示，对​​于某些s <= 5，每个立方有限对称立方图都是s-正则的。我们显示，最多9个围的对称立方图是I-正则或2'-正则（遵循德约科维奇（Djokovic）的符号），或属于一小类特殊图形。另一方面，我们显示出有无限多的3个正则三次的围长10的三次方图，因此最多9个围长的陈述不能改进为较大围长的立方图。我们还给出了周长g <= 9的1-或2'-正则立方图的特征，证明除了五个例外，它们在每种情况下都与三角形群Delta（2，3，g）的商密切相关，或在g = 8的情况下属于（x，y竖线x（2）= y（3）= [x，y]（4）= 1）中的所有或所有3个传递立方图以及特殊的1-和由Conder和Dobcsanyi产生的多达768个顶点的立方对称图列表中，最多有9个周长的2个正则立方图[M. Conder，P。Dobcsanyi，多达768个顶点的三价对称图，J。Combin。数学。组合计算40（2002）41-63];最大的是570阶（和周长9）的3正则图F570。主要结果的证明是计算机辅助的。 （c）2007 Elsevier Inc.保留所有权利。

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