Beyond symmetry in generalized Petersen graphs

TL;DR

This paper characterizes generalized Petersen graphs that are cores, explores their endomorphism-transitivity, and investigates which are Cayley graphs of monoids, extending previous classifications and solving open problems.

Contribution

It provides a complete characterization of core generalized Petersen graphs and identifies those that are Cayley graphs of monoids, extending prior work and solving open questions.

Findings

Characterized when generalized Petersen graphs are cores.

Identified generalized Petersen graphs that are Cayley graphs of monoids.

Extended existing classifications of vertex- and group-transitive generalized Petersen graphs.

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 graph of a monoid with generating connection set of size two. This extends Nedela and \v{S}koviera's characterization of generalized Petersen graphs that are group Cayley graphs and complements results of Hao, Gao, and Luo.