2022/02/14 by Ignacio García-Marco, García-Marco, Ignacio, Kolja Knauer +1 · 3 citations
Computer Science · Mathematics · #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #cs.DM #math.CO
paper · pdf · doi:10.48550/arxiv.2202.06785
22 pages, 13 figures, 2 tables
arxiv created 2022/02/14 · openalex publication_date 2022/02/14 · arxiv updated 2022/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.