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Please use this identifier to cite or link to this item: https://hdl.handle.net/2445/217280
Beyond symmetry in generalized Petersen graphs
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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 vertextransitive 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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GARCÍA MARCO, Ignacio and KNAUER, Kolja. Beyond symmetry in generalized Petersen graphs. Journal of Algebraic Combinatorics. 2024. Vol. 59, num. 2, pags. 331-357. ISSN 0925-9899. [consulted: 15 of June of 2026]. Available at: https://hdl.handle.net/2445/217280