Download Wolfram Notebook

A pretzel graph is a graph with graph genus 3 (West 2000, p. 266). Planar, toroidal graphs, and double-toroidal graphs are therefore not pretzel in this exact-genus terminology, even though they can be embedded on a genus-3 surface.

Examples of pretzel graphs include the complete graph and complete bipartite graph . Other nice named examples include 10-cocktail party graph, 12-crown graph, Hesse graph, Folkman graph, Coxeter graph, and Schläfli double six graph.

There are no pretzel graphs on 8 or fewer vertices, and the 12 pretzel graphs on 9 vertices (E. Weisstein, Jul. 2, 2026) are illustrated above.

See also

Double-Toroidal Graph, Graph Genus, Planar Graph, Toroidal Graph

Explore with Wolfram|Alpha

References

West, D. B. "Surfaces of Higher Genus." Introduction to Graph Theory, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 266-269, 2000.

Referenced on Wolfram|Alpha

Pretzel Graph

Cite this as:

Weisstein, Eric W. "Pretzel Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PretzelGraph.html

Subject classifications