Herschel graph

English

The Herschel graph

Etymology

From Herschel (“a surname”) + graph, after British astronomer Alexander Stewart Herschel (1836—1907), who identified the associated polyhedron (an enneahedron) as one for which there is no solution to the icosian game.

Pronunciation

• Rhymes: -æf

Proper noun

Herschel graph

1. (mathematics, graph theory) A bipartite undirected graph with 11 vertices and 18 edges that is the smallest non-Hamiltonian polyhedral graph.
   • 1994, Ralph P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction‎^[1], page 587:

     Select a suitable independent set / and use part (b) to show that the graph in Fig. 11.81 (known as the Herschel graph) has no Hamilton cycle.

   • 2004, William Kocay, Donald L. Kreher, Graphs, Algorithms, and Optimization, page 202:

     A bipartite graph like the Herschel graph of Figure 9.2 is also non-hamiltonian, but the algorithm is not likely to delete enough vertices to notice that it has a large separating set.

   • 2006, Michael S. Keane, Dee Denteneer, Frank Hollander, Evgeny Verbitskiy, Dynamics and Stochastics, Institute of Mathematical Statistics, Lecture Notes—Monograph Series, Volume 48, page 174,

     It is difficult to control what loops may arise: for example the Herschel graph [3] shows that a convex polyhedron need not be Hamiltonian as a graph.

Synonyms

• (smallest non-Hamiltonian polyhedral graph): Herschel's graph

See also

• icosian game on Wikipedia.Wikipedia