| Herschel graph | |
|---|---|
 The Herschel graph. | |
| Named after | Alexander Stewart Herschel |
| Vertices | 11 |
| Edges | 18 |
| Automorphisms | 12 (D6) |
| Properties | |
| Table of graphs and parameters | |
Ingraph theory, a branch ofmathematics, theHerschel graph is abipartiteundirected graph with 11 vertices and 18 edges. It is apolyhedral graph (the graph of aconvex polyhedron), and is the smallest polyhedral graph that does not have aHamiltonian cycle, a cycle passing through all its vertices. The polyhedron whose vertices and edges form this graph is sometimes called theHerschel enneahedron; it is anenneahedron because it has nine faces.
Both the graph and the polyhedron are named after British astronomerAlexander Stewart Herschel, because of Herschel's studies of Hamiltonian cycles in polyhedral graphs (but not of this graph).
The Herschel graph has three vertices of degree four (the three blue vertices aligned vertically in the center of the illustration) and eight vertices of degree three. Each two distinct degree-four vertices share two degree-three neighbors, forming a four-vertex cycle with these shared neighbors. There are three of these cycles, passing through six of the eight degree-three vertices (red in the illustration). Two more degree-three vertices (blue) do not participate in these four-vertex cycles; instead, each is adjacent to three of the six red vertices.[1]
The Herschel graph is apolyhedral graph; this means that it is aplanar graph, one that can be drawn in the plane with none of its edges crossing, and that it is3-vertex-connected: the removal of any two of its vertices leaves a connectedsubgraph.[1] It is abipartite graph: when it is colored with five blue and six red vertices, as illustrated, each edge has one red endpoint and one blue endpoint.[2]
It has order-6dihedral symmetry, for a total of 12 members of itsautomorphism group. The degree-four vertices can be permuted arbitrarily, giving six permutations, and in addition, for each permutation of the degree-four vertices, there is a symmetry that keeps these vertices fixed and exchanges pairs of degree-three vertices.[1]
BySteinitz's theorem, every graph that is planar and 3-vertex-connected is theskeleton of someconvex polyhedron.[3] Because the Herschel graph has these properties,[1] it can be represented in this way by a convexenneahedron, a polyhedron whose faces are nine quadrilaterals.[4] This can be designed so that eachgraph automorphism corresponds to a symmetry of the polyhedron, in which case three of the faces will berhombi or squares, and the other six will bekites.[1]
Thedual polyhedron is arectified triangular prism, which can be formed as theconvex hull of the midpoints of the edges of atriangular prism. When constructed in this way, it has three square faces on the same planes as the square faces of the prism, twoequilateral triangle faces on the planes of the triangular ends of the prism, and six moreisosceles triangle faces. This polyhedron has the property that its faces cannot be numbered in such a way that consecutive numbers appear on adjacent faces, and such that the first and last numbers are also on adjacent faces, because such a numbering would necessarily correspond to a Hamiltonian cycle in the Herschel graph. Polyhedral face numberings of this type are used as "spindown life counters" in the gameMagic: The Gathering, to track player lives, by turning the polyhedron to an adjacent face whenever a life is lost. A card in the game, the Lich, allows players to return from a nearly-lost state with a single life to their initial number of lives. Because the dual polyhedron for the Herschel graph cannot be numbered in such a way that this step connects adjacent faces,Constantinides & Constantinides (2018) name thecanonical polyhedron realization of this dual polyhedron as "the Lich's nemesis".[5]
As a bipartite graph that has an odd number of vertices, the Herschel graph does not contain aHamiltonian cycle (a cycle of edges that passes through each vertex exactly once). For, in any bipartite graph, any cycle must alternate between the vertices on either side of the bipartition, and therefore must contain equal numbers of both types of vertex and must have an even length. Thus, a cycle passing once through each of the eleven vertices cannot exist in the Herschel graph. A graph is called Hamiltonian whenever it contains a Hamiltonian cycle, so the Herschel graph is not Hamiltonian. It has the smallest number of vertices, the smallest number of edges, and the smallest number of faces of any non-Hamiltonian polyhedral graph.[6] There exist other polyhedral graphs with 11 vertices and no Hamiltonian cycles (notably theGoldner–Harary graph)[7] but none with fewer edges.[6]
All but three of the vertices of the Herschel graph have degree three. A graph is calledcubic or3-regular when all of its vertices have degree three.P. G. Tait conjectured[8] that a polyhedral 3-regular graph must be Hamiltonian; this was disproved whenW. T. Tutte provided a counterexample, theTutte graph, which is much larger than the Herschel graph.[9] A refinement of Tait's conjecture,Barnette's conjecture that every bipartite 3-regular polyhedral graph is Hamiltonian, remains open.[10]
Everymaximal planar graph that does not have a Hamiltonian cycle has a Herschel graph as aminor. The Herschel graph is conjectured to be one of three minor-minimal non-Hamiltonian 3-vertex-connected graphs. The other two are thecomplete bipartite graph and a graph formed by splitting both the Herschel graph and into two symmetric halves by three-vertex separators and then combining one half from each graph.[11]
The Herschel graph also provides an example of a polyhedral graph for which themedial graph has noHamiltonian decomposition into two edge-disjoint Hamiltonian cycles. The medial graph of the Herschel graph is a 4-regular graph with 18 vertices, one for each edge of the Herschel graph; two vertices are adjacent in the medial graph whenever the corresponding edges of the Herschel graph are consecutive on one of its faces.[12] It is4-vertex-connected andessentially6-edge-connected. Here, a graph is-vertex-connected or-edge-connected if the removal of fewer than vertices or edges (respectively) cannot disconnected it. Planar graphs cannot be 6-edge-connected, because they always have a vertex of degree at most five, and removing the neighboring edges disconnects the graph. The "essentially 6-edge-connected" terminology means that this trivial way of disconnecting the graph is ignored, and it is impossible to disconnect the graph into two subgraphs that each have at least two vertices by removing five or fewer edges.[13]
The Herschel graph is named afterAlexander Stewart Herschel, a British astronomer, who wrote an early paper concerningWilliam Rowan Hamilton'sicosian game. This is a puzzle involving finding Hamiltonian cycles on a polyhedron, usually theregular dodecahedron. The Herschel graph describes the smallestconvex polyhedron that can be used in place of the dodecahedron to give a game that has no solution. Herschel's paper described solutions for the Icosian game only on the graphs of theregular tetrahedron andregular icosahedron; it did not describe the Herschel graph.[14] The name "the Herschel graph" makes an early appearance in a graph theory textbook byJohn Adrian Bondy andU. S. R. Murty, published in 1976.[15] The graph itself was described earlier, for instance byH. S. M. Coxeter.[4]
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