Goldner–Harary graph

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Goldner–Harary graph
Goldner-Harary graph.svg
Named after A. Goldner,
Frank Harary
Vertices 11
Edges 27
Radius 2
Diameter 2
Girth 3
Automorphisms 12 (D6)
Chromatic number 4
Chromatic index 8
Properties Polyhedral
Treewidth 3

In the mathematical field of graph theory, the Goldner–Harary graph is a simple undirected graph with 11 vertices and 27 edges. It is named after A. Goldner and Frank Harary, who proved in 1975 that it was the smallest non-Hamiltonian maximal planar graph.[1][2][3] The same graph had already been given as an example of a non-Hamiltonian simplicial polyhedron by Branko Grünbaum in 1967.[4]


The Goldner–Harary graph is a planar graph: it can be drawn in the plane with none of its edges crossing. When drawn on a plane, all its faces are triangular, making it a maximal planar graph. As with every maximal planar graph, it is also 3-vertex-connected: the removal of any two of its vertices leaves a connected subgraph.

The Goldner–Harary graph is also non-hamiltonian. The smallest possible number of vertices for a non-hamiltonian polyhedral graph is 11. Therefore, the Goldner–Harary graph is a minimal example of graphs of this type. However, the Herschel graph, another non-Hamiltonian polyhedron with 11 vertices, has fewer edges.

As a non-Hamiltonian maximal planar graph, the Goldner–Harary graph provides an example of a planar graph with book thickness greater than two.[5] Based on the existence of such examples, Bernhart and Kainen conjectured that the book thickness of planar graphs could be made arbitrarily large, but it was subsequently shown that all planar graphs have book thickness at most four.[6]

It has book thickness 3, chromatic number 4, chromatic index 8, girth 3, radius 2, diameter 2 and is a 3-edge-connected graph.

It is also a 3-tree, and therefore it has treewidth 3. Like any k-tree, it is a chordal graph. As a planar 3-tree, it forms an example of an Apollonian network.


By Steinitz's theorem, the Goldner–Harary graph is a polyhedral graph: it is planar and 3-connected, so there exists a convex polyhedron having the Goldner–Harary graph as its skeleton.

Geometrically, a polyhedron representing the Goldner–Harary graph may be formed by gluing a tetrahedron onto each face of a triangular dipyramid, similarly to the way a triakis octahedron is formed by gluing a tetrahedron onto each face of an octahedron. That is, it is the Kleetope of the triangular dipyramid.[4][7] The dual graph of the Goldner–Harary graph is represented geometrically by the truncation of the triangular prism.

Algebraic properties[edit]

The automorphism group of the Goldner–Harary graph is of order 12 and is isomorphic to the dihedral group D6, the group of symmetries of a regular hexagon, including both rotations and reflections.

The characteristic polynomial of the Goldner–Harary graph is : -(x-1)^2 x^2 (x+2)^3 (x^2-3) (x^2-4 x-9).


  1. ^ Goldner, A.; Harary, F. (1975), "Note on a smallest nonhamiltonian maximal planar graph", Bull. Malaysian Math. Soc. 6 (1): 41–42 . See also the same journal 6(2):33 (1975) and 8:104-106 (1977). Reference from listing of Harary's publications.
  2. ^ Dillencourt, M. B. (1996), "Polyhedra of small orders and their Hamiltonian properties", Journal of Combinatorial Theory, Series B 66: 87–122, doi:10.1006/jctb.1996.0008 .
  3. ^ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford, England: Oxford University Press, p. 285 .
  4. ^ a b Grünbaum, Branko (1967), Convex Polytopes, Wiley Interscience, p. 357 . Same page, 2nd ed., Graduate Texts in Mathematics 221, Springer-Verlag, 2003, ISBN 978-0-387-40409-7.
  5. ^ Bernhart, Frank R.; Kainen, Paul C. (1979), "The book thickness of a graph", Journal of Combinatorial Theory, Series B 27 (3): 320–331, doi:10.1016/0095-8956(79)90021-2  |chapter= ignored (help).
  6. ^ Yannakakis, Mihalis (1986), "Four pages are necessary and sufficient for planar graphs", Proc. 18th ACM Symp. Theory of Computing (STOC), pp. 104–108, doi:10.1145/12130.12141 .
  7. ^ Ewald, Günter (1973), "Hamiltonian circuits in simplicial complexes", Geometriae Dedicata 2 (1): 115–125, doi:10.1007/BF00149287 .

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