Beyond symmetry in generalized Petersen graphs

IF 0.6 3区 数学 Q3 MATHEMATICS Journal of Algebraic Combinatorics Pub Date : 2024-01-24 DOI:10.1007/s10801-023-01282-y
Ignacio García-Marco, Kolja Knauer
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Abstract

A graph is a core or unretractive if all its endomorphisms are automorphisms. Well-known examples of cores include the Petersen graph and the graph of the dodecahedron—both generalized Petersen graphs. We characterize the generalized Petersen graphs that are cores. A simple characterization of endomorphism-transitive generalized Petersen graphs follows. This extends the characterization of vertex-transitive generalized Petersen graphs due to Frucht, Graver, and Watkins and solves a problem of Fan and Xie. Moreover, we study generalized Petersen graphs that are (underlying graphs of) Cayley graphs of monoids. We show that this is the case for the Petersen graph, answering a recent mathoverflow question, for the Desargues graphs, and for the Dodecahedron—answering a question of Knauer and Knauer. Moreover, we characterize the infinite family of generalized Petersen graphs that are Cayley graphs of a monoid with generating connection set of size two. This extends Nedela and Škoviera’s characterization of generalized Petersen graphs that are group Cayley graphs and complements results of Hao, Gao, and Luo.

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超越广义彼得森图的对称性
如果一个图的所有内变形都是自动变形,那么这个图就是核心图或不可延展图。核心图的著名例子包括彼得森图和十二面体图--它们都是广义彼得森图。我们将描述作为核的广义彼得森图的特征。接下来是内态性广义彼得森图的简单表征。这扩展了由 Frucht、Graver 和 Watkins 提出的顶点传递广义彼得森图的表征,并解决了 Fan 和 Xie 提出的一个问题。此外,我们研究的广义彼得森图是单体的 Cayley 图的(底层图)。我们证明了彼得森图、德萨格图和十二面体都是这种情况,回答了最近 mathoverflow 提出的一个问题。此外,我们还描述了广义彼得森图的无穷族,它们是大小为 2 的生成连接集的单元的 Cayley 图。这扩展了内德拉和什科维耶拉对广义彼得森图的表征,这些广义彼得森图是组 Cayley 图,并补充了郝、高和罗的结果。
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来源期刊
CiteScore
1.50
自引率
12.50%
发文量
94
审稿时长
6-12 weeks
期刊介绍: The Journal of Algebraic Combinatorics provides a single forum for papers on algebraic combinatorics which, at present, are distributed throughout a number of journals. Within the last decade or so, algebraic combinatorics has evolved into a mature, established and identifiable area of mathematics. Research contributions in the field are increasingly seen to have substantial links with other areas of mathematics. The journal publishes papers in which combinatorics and algebra interact in a significant and interesting fashion. This interaction might occur through the study of combinatorial structures using algebraic methods, or the application of combinatorial methods to algebraic problems.
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