论文标题
超越对称性的彼得森图
Beyond symmetry in generalized Petersen graphs
论文作者
论文摘要
如果其所有内态都是自动形态的，则图是核心或非归因性。核心的众所周知的例子包括彼得森图和十二面体的图 - 均广义彼得森图。我们表征了核心的广义彼得森图。随之而来的是内态递增的普遍彼得森图的简单表征。这扩展了由于Frucht，Graver和Watkins引起的顶点传播的广义彼得森图的表征，并解决了风扇和XIE的问题。 此外，我们研究了Monoids的Cayley图的广义彼得森图。我们表明，彼得森图是这种情况，回答了最近的Mathoverflow问题，Desargues图和十二面体 - 回答了Knauer和Knauer的问题。此外，我们表征了无限的广义彼得森图家族，这些家族是单型的Cayley图，并产生了尺寸二的连接集。这扩展了Nedela和škoviera对cayley图组的广义彼得森图的描述，并补充了Hao，Gao和Luo的结果。
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 graph 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.
