2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/32523We consider the class of the topologically locally finite (in short TLF) planar vertex-transitive graphs, a class containing in particular all the one-ended planar Cayley graphs and the normal transitive tilings. We characterize these graphs with a finite local representation and a special kind of finite state automaton named labeling scheme. As a result, we are able to enumerate and describe all TLF-planar vertex-transitive graphs of any given degree. Also, we are able decide to whether any TLF-planar transitive graph is Cayley or not.Article : 23 pages, 15 figures Appendix : 13 pages, 72 figures Submitted to Discrete Mathematics The appendix is accessible at http://www.labri.fr/~renault/research/research.htmlDiscrete MathematicsG.2.1; G.2.2text