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| name = Heawood graph
| image = [[Image:Heawood_Graph.svg|180px]]
| image_caption =
| namesake = [[Percy John Heawood]]
| vertices = 14
| edges = 21
| girth = 6
| diameter = 3
| radius = 3
| automorphisms = 336 ([[Projective linear group|PGL<sub>2</sub>(7)]])
| chromatic_number = 2
| chromatic_index = 3
| properties = [[Bipartite graph|Bipartite]]<br>[[Cubic graph|Cubic]]<br>[[Cage (graph theory)|Cage]]<br>[[Distance-transitive graph|Distance-transitive]]<br>[[Distance-regular graph|Distance-regular]]<br>[[Toroidal graph|Toroidal]]<br>[[Hamiltonian graph|Hamiltonian]]<br>[[Symmetric graph|Symmetric]]<br>Orientably simple
}}


In the [[mathematics|mathematical]] field of [[graph theory]], the '''Heawood graph''' is an [[undirected graph]] with 14 vertices and 21 edges, named after [[Percy John Heawood]].<ref>{{MathWorld | urlname=HeawoodGraph | title=Heawood Graph}}</ref>
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==Combinatorial properties==
The graph is [[cubic graph|cubic]], and all cycles in the graph have six or more edges. Every smaller cubic graph has shorter cycles, so this graph is the 6-[[cage (graph theory)|cage]], the smallest cubic graph of [[girth (graph theory)|girth]] 6. It is a [[distance-transitive graph]] (see the [[Foster census]]) and therefore [[Distance-regular graph|distance regular]].<ref name="brouwer">{{cite web | author=Brouwer, Andries E. | authorlink = Andries Brouwer | title = Heawood graph | url = http://www.win.tue.nl/~aeb/drg/graphs/Heawood.html | work = [http://www.win.tue.nl/~aeb/drg/index.html Additions and Corrections to the book ''Distance-Regular Graphs''] (Brouwer, Cohen, Neumaier; Springer; 1989)}}</ref>
 
There are 24 [[perfect matching]]s in the Heawood graph; for each matching, the set of edges not in the matching forms a [[Hamiltonian cycle]]. For instance, the figure shows the vertices of the graph placed on a cycle, with the internal diagonals of the cycle forming a matching. By subdividing the cycle edges into two matchings, we can partition the Heawood graph into three perfect matchings (that is, [[Edge coloring|3-color its edges]]) in eight different ways.<ref name="brouwer"/> Every two perfect matchings, and every two Hamiltonian cycles, can be transformed into each other by a symmetry of the graph.<ref>{{citation
| last1 = Abreu | first1 = M.
| last2 = Aldred | first2 = R. E. L.
| last3 = Funk | first3 = M.
| last4 = Jackson | first4 = Bill
| last5 = Labbate | first5 = D.
| last6 = Sheehan | first6 = J.
| doi = 10.1016/j.jctb.2004.09.004
| mr = 2099150
| issue = 2
| journal = [[Journal of Combinatorial Theory]] | series = Series B
| pages = 395–404
| title = Graphs and digraphs with all 2-factors isomorphic
| volume = 92
| year = 2004}}.</ref>
 
There are 28 six-vertex cycles in the Heawood graph. Each 6-cycle is disjoint from exactly three other 6-cycles; among these three 6-cycles, each one is the symmetric difference of the other two. The graph with one node per 6-cycle, and one edge for each disjoint pair of 6-cycles, is the [[Coxeter graph]].<ref>{{citation|first=Italo J.|last=Dejter|title=From the Coxeter graph to the Klein graph|journal=Journal of Graph Theory|year=2011|doi=10.1002/jgt.20597|arxiv=1002.1960}}.</ref>
 
==Geometric and topological properties==
The Heawood graph is a [[toroidal graph]]; that is, it can be embedded without crossings onto a [[torus]]. One embedding of this type places its vertices and edges into three-dimensional [[Euclidean space]] as the set of vertices and edges of a nonconvex polyhedron with the topology of a torus, the [[Szilassi polyhedron]].  
 
The graph is named after [[Percy John Heawood]], who in 1890 proved that in every subdivision of the torus into polygons, the polygonal regions can be colored by at most seven colors.<ref>{{cite journal | doi=10.2307/3219140 | author=Brown, Ezra | title = The many names of (7,3,1) | journal = [[Mathematics Magazine]] | volume=75 | issue=2 | year=2002 | pages=83–94 | url=http://www.math.vt.edu/people/brown/doc/731.pdf | jstor=3219140}}</ref><ref>{{cite journal | author=Heawood, P. J. | authorlink=Percy John Heawood | title=Map colouring theorems | journal = Quarterly J. Math. Oxford Ser. | year=1890 | volume=24 | pages=322–339}}</ref> The Heawood graph forms a subdivision of the torus with seven mutually adjacent regions, showing that this bound is tight.
 
The Heawood graph is also the [[Levi graph]] of the [[Fano plane]], the graph representing incidences between points and lines in that geometry. With this interpretation, the 6-cycles in the Heawood graph correspond to [[triangle]]s in the Fano plane.
 
The Heawood graph has [[Crossing number (graph theory)|crossing number]] 3, and is the smallest cubic graph with that crossing number {{OEIS|id=A110507}}. Including the Heawood graph, there are 8 distinct graphs of order 14 with crossing number 3.
 
The Heawood graph is a [[unit distance graph]]: it can be embedded in the plane such that adjacent vertices are exactly at distance one apart, with no two vertices embedded to the same point and no vertex embedded into a point within an edge. However, the known embeddings of this type lack any of the symmetries that are inherent in the graph.<ref>{{Cite journal | last = Gerbracht | first = Eberhard H.-A. | title = Eleven unit distance embeddings of the Heawood graph | year = 2009 | arxiv = 0912.5395}}.</ref>
 
==Algebraic properties==
The [[automorphism group]] of the Heawood graph is isomorphic to the [[projective linear group]] PGL<sub>2</sub>(7), a group of order 336.<ref>{{cite book|last1=Bondy|first1=J. A.|author1-link=John Adrian Bondy|last2=Murty|first2=U. S. R.|author2-link=U. S. R. Murty|title=Graph Theory with Applications|location=New York|publisher=North Holland|year=1976|page=237|isbn=0-444-19451-7|url=http://www.ecp6.jussieu.fr/pageperso/bondy/books/gtwa/gtwa.html}}</ref> It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Heawood graph is a [[symmetric graph]]. It has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the [[Foster census]], the Heawood graph, referenced as F014A, is the only cubic symmetric graph on 14 vertices.<ref>Royle, G. [http://www.cs.uwa.edu.au/~gordon/remote/foster/#census "Cubic Symmetric Graphs (The Foster Census)."]</ref><ref>[[Marston Conder|Conder, M.]] and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41-63, 2002.</ref>
 
The [[characteristic polynomial]] of the Heawood graph is <math>(x-3) (x+3) (x^2-2)^6</math>. It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.
 
==Gallery==
<gallery>
Image:Szilassi polyhedron.svg|The [[Szilassi polyhedron]].
File:3-crossing Heawood graph.svg|The Heawood graph has [[Crossing number (graph theory)|crossing number]]&nbsp;3.
File:Heawood graph 3color edge.svg|The [[chromatic index]] of the Heawood graph is&nbsp;3.
File:Heawood graph 2COL.svg|The [[chromatic number]] of the Heawood graph is&nbsp;2.
File:7x-torus.svg|An embedding of the Heawood graph onto a torus (shown as a square with [[periodic boundary conditions]]) partitioning it into seven mutually-adjacent regions
</gallery>
 
== References ==
{{reflist}}
 
[[Category:Individual graphs]]
[[Category:Regular graphs]]

Latest revision as of 17:03, 6 April 2014

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