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Concept
Grötzsch graph
Summary
In the mathematical field of graph theory, the Grötzsch graph is a triangle-free graph with 11 vertices, 20 edges, chromatic number 4, and crossing number 5. 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.
The Grötzsch graph is a member of an infinite sequence of triangle-free graphs, each the Mycielskian of the previous graph in the sequence, starting from the one-edge graph; this sequence of graphs was constructed by to show that there exist triangle-free graphs with arbitrarily large chromatic number. Therefore, the Grötzsch graph is sometimes also called the Mycielski graph or the Mycielski–Grötzsch graph. Unlike later graphs in this sequence, the Grötzsch graph is the smallest triangle-free graph with its chromatic number.
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. 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.
The characteristic polynomial of the Grötzsch graph is
Although it is not a planar graph, it can be embedded in the projective plane without crossings. This embedding has ten faces, all of which are quadrilaterals.
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. It has odd girth five but girth four, and does not have any graph homomorphism to a graph whose girth is five or more, so it forms an example that distinguishes odd girth from the maximum girth that can be obtained from a homomorphism.
used a modified version of the Grötzsch graph to disprove a conjecture of on the chromatic number of triangle-free graphs with high degree.
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