It is named after German mathematician Herbert Grötzsch, who used it as an example in connection with his 1959 theorem that planar triangle-free graphs are 3-colorable.

[2] The full automorphism group of the Grötzsch graph is isomorphic to the dihedral group D5 of order 10, the group of symmetries of a regular pentagon, including both rotations and reflections.

[3] These symmetries have three orbits of vertices: the degree-5 vertex (by itself), its five neighbors, and its five non-neighbors.

Similarly, there are three orbits of edges, distinguished by their distance from the degree-5 vertex.

Although it is not a planar graph, it can be embedded in the projective plane without crossings.

[4] The existence of the Grötzsch graph demonstrates that the assumption of planarity is necessary in Grötzsch's theorem that every triangle-free planar graph is 3-colorable.

[5] Häggkvist (1981) used a modified version of the Grötzsch graph to disprove a conjecture of Paul Erdős and Miklos Simonovits (1973) on the chromatic number of triangle-free graphs with high degree.

Two vertices in this expanded graph are connected by an edge if they correspond to vertices connected by an edge in the Grötzsch graph.

The result of Häggkvist's construction is a 10-regular triangle-free graph with 29 vertices and chromatic number 4, disproving the conjecture that there is no 4-chromatic triangle-free