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[[File:Grinberg 5CEC Nonhamiltonian graph.svg|thumb|A graph that can be proven non-Hamiltonian using Grinberg's theorem]]
In [[graph theory]], '''Grinberg's theorem''' is a necessary condition for a [[planar graph]] to contain a [[Hamiltonian cycle]], based on the lengths of its face cycles.  The result has been widely used to construct non-Hamiltonian planar graphs with further properties, such as to give new [[counterexample]]s to [[Tait's conjecture]] (originally disproved by [[W.T. Tutte]] in 1946).  This theorem was proved by [[Latvia]]n mathematician [[Emanuel Grinberg]] in 1968.


== Precise formulation ==
Let ''G'' be a finite plane graph with a Hamiltonian cycle ''C''. 
Denote by ''&fnof;''<sub>''k''</sub> and ''g''<sub>''k''</sub> the number of ''k''-gonal faces inside and outside of ''C'', respectively.  Then


: <math>\sum_{k \geq 3} (k-2) (f_k-g_k) = 0.</math>
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The proof is an easy consequence of [[Euler characteristic|Euler's formula]].
 
==Applications==
Grinberg used his theorem to find non-Hamiltonian [[cubic graph|cubic]] [[polyhedral graph]]s with high cyclic edge connectivity. The cyclic edge connectivity of a graph is the smallest number of edges that may be deleted in such a way that the remaining graph has more than one cyclic component. The 46-vertex [[Tutte graph]], and the smaller cubic non-Hamiltonian polyhedral graphs derived from it, have cyclic edge connectivity three. Grinberg used his theorem to find a non-Hamiltonian cubic polyhedral graph with 44 vertices, 24 faces, and cyclic edge connectivity four, and another example (shown in the figure) with 46 vertices, 25 faces, and cyclic edge connectivity five, the maximum possible cyclic edge connectivity for a cubic planar graph other than ''K''<sub>4</sub/>. In the example shown, all of the bounded faces have either five or eight edges, and therefore (because of the factor of ''k''&nbsp;&minus;&nbsp;2 in the sum) they contribute zero modulo 3 to the sum in Grinberg's theorem, regardless of whether they are inside or outside of any Hamiltonian cycle ''C''. The unbounded face, however, has nine edges, and contributes a nonzero term modulo three to the sum. Therefore, the overall summation has a value that is nonzero modulo three, and Grinberg's theorem implies that the graph cannot be Hamiltonian.
 
Grinberg's theorem has also been used to find planar [[hypohamiltonian graph]]s, again by making all but one face have a number of edges congruent to 2 mod 3 ({{harvnb|Thomassen|1976}}, {{harvnb|Wiener|Araya|2009}}). {{harvtxt|Thomassen|1981}} uses the theorem in a somewhat more complicated way to find a planar [[cubic graph|cubic]] hypohamiltonian graph: the graph he constructs includes a 4-edge face adjacent to four 7-edge faces, and all other faces have five or eight edges. In order to satisfy Grinberg's theorem, a Hamiltonian cycle would have to separate one of the 4- or 7-edge faces from the other four, which is not possible.
 
==Limitations==
There exist planar non-Hamiltonian graphs in which all faces have five or eight sides. For these graphs, Grinberg's formula taken modulo three is always satisfied by any partition of the faces into two subsets, preventing the application of his theorem to proving non-Hamiltonicity in this case {{harv|Zaks|1977}}.
 
It is not possible to use Grinberg's theorem to find counterexamples to [[Barnette's conjecture]], that every cubic [[bipartite graph|bipartite]] [[polyhedral graph]] is Hamiltonian. For, in such graphs, there always exists a partition of the faces into two subsets satisfying Grinberg's theorem, regardless of Hamiltonicity {{harv|Krooss|2004}}.
 
== References ==
*{{citation
| last = Grinberg | first = È. Ja. | authorlink = Emanuels Grīnbergs
| contribution = Plane homogeneous graphs of degree three without Hamiltonian circuits
| language = Russian
| location = Riga
| mr = 0238732
| pages = 51–58
| publisher = Izdat. “Zinatne”
| title = Latvian Math. Yearbook 4
| year = 1968}}. English translation by Dainis Zeps, {{arxiv|0908.2563}}.
*{{citation
| last = Krooss | first = André
| journal = Analele Universităţii din Craiova. Seria Matematică-Informatică
| language = German
| mr = 2153849
| pages = 59–65
| title = Die Barnette'sche Vermutung und die Grinberg'sche Formel
| volume = 31
| year = 2004}}.
*{{citation
| last = Malkevitch | first = Joseph
| date = January 2005
| publisher = [[American Mathematical Society]]
| series = Feature Column
| title = Euler's Polyhedral Formula: Part II
| url = http://www.ams.org/featurecolumn/archive/eulers-formulaII.html}}.
*{{citation
| last = Thomassen | first = Carsten | authorlink = Carsten Thomassen
| title = Planar and infinite hypohamiltonian and hypotraceable graphs
| journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
| volume = 14 | year = 1976 | pages = 377–389
| id = {{MathSciNet | id = 0422086}}
| doi = 10.1016/0012-365X(76)90071-6
| issue = 4}}.
*{{citation
| last = Thomassen | first = Carsten | authorlink = Carsten Thomassen
| doi = 10.1016/0095-8956(81)90089-7
| issue = 1
| journal = [[Journal of Combinatorial Theory]] | series = Series B
| mr = 609592
| pages = 36–44
| title = Planar cubic hypo-Hamiltonian and hypotraceable graphs
| volume = 30
| year = 1981}}.
*{{citation
| last1 = Wiener | first1 = Gábor
| last2 = Araya | first2 = Makoto
| title = The ultimate question
| year = 2009
| arxiv = 0904.3012}}.
*{{citation
| last = Tutte | first = W. T. | authorlink = W. T. Tutte
| contribution = Chapter 2: Knights Errant
| isbn = 0-19-850251-6
| location = New York
| mr = 1635397
| publisher = The Clarendon Press Oxford University Press
| series = Oxford Lecture Series in Mathematics and its Applications
| title = Graph theory as I have known it
| volume = 11
| year = 1998}}.
*{{citation
| last = Zaks | first = Joseph
| doi = 10.1016/0012-365X(77)90165-0
| issue = 3
| journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
| mr = 0460189
| pages = 317–321
| title = Non-Hamiltonian non-Grinbergian graphs
| volume = 17
| year = 1977}}.
 
== External links ==
* [http://mathworld.wolfram.com/GrinbergGraphs.html Grinberg Graphs], from ''[[MathWorld]]''.
 
[[Category:Theorems in graph theory]]
[[Category:Planar graphs]]
[[Category:Hamiltonian paths and cycles]]

Latest revision as of 01:27, 25 September 2014


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