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Keywords:

  • automorphism group;
  • bipartite graph;
  • Heawood and co-Heawood graph;
  • Coxeter graph;
  • Fano plane;
  • girth of a graph;
  • (hyperbolic) honeycomb;
  • Klein map inline image-regular graph;
  • locally symmetric graph;
  • matching;
  • 120-cell;
  • Petersen graph;
  • regular dodecahedron;
  • Riemann surface;
  • tessellation;
  • bathroom tiling;
  • MSC (2000) :05C12, 05C30, 51E30, 94C15

Abstract

We classify the family of connected, locally symmetric graphs of girth 4 (finite and infinite). They are all regular, with the exception of the complete bipartite graph inline image. There are, up to isomorphism, exactly four such k-regular graphs for every inline image, one for inline image, two for inline image, and exactly three for every infinite cardinal k. In the last paragraph, we consider locally symmetric graphs of girth >4.