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.
