Omnitruncated 6-simplex honeycomb: Difference between revisions

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In [[graph theory]], an '''acyclic orientation''' of an [[undirected graph]] is an assignment of a direction to each edge (an [[Orientation (graph theory)|orientation]]) that does not form any [[directed cycle]] and therefore makes it into a [[directed acyclic graph]]. Every graph has an acyclic orientation; all acyclic orientations may be obtained by placing the vertices into a sequence, and then directing each edge from the earlier of its endpoints in the sequence to the later endpoint.
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The [[Gallai–Hasse–Roy–Vitaver theorem]] states that a graph has an acyclic orientation in which the [[longest path]] has ''k'' vertices if and only if it can be [[graph coloring|colored]] with ''k'' colors.<ref>{{citation
| last1 = Nešetřil | first1 = Jaroslav | author1-link = Jaroslav Nešetřil
| last2 = Ossona de Mendez | first2 = Patrice
| contribution = Theorem 3.13
| doi = 10.1007/978-3-642-27875-4
| isbn = 978-3-642-27874-7
| location = Heidelberg
| mr = 2920058
| page = 42
| publisher = Springer
| series = Algorithms and Combinatorics
| title = Sparsity: Graphs, Structures, and Algorithms
| volume = 28
| year = 2012}}.</ref>
 
For [[planar graph]]s, acyclic orientations are dual to [[strong orientation|totally cyclic orientation]]s, orientations in which each edge belongs to a directed cycle: if <math>G</math> is a [[planar graph]], and orientations of <math>G</math> are transferred to orientations of the [[planar dual]] graph of <math>G</math> by turning each edge 90 degrees clockwise, then a totally cyclic orientation of <math>G</math> corresponds in this way to an acyclic orientation of the dual graph and vice versa.<ref>{{citation
| last = Welsh | first = Dominic
| contribution = Approximate counting
| doi = 10.1017/CBO9780511662119.010
| location = Cambridge
| mr = 1477750
| pages = 287–323
| publisher = Cambridge Univ. Press
| series = London Math. Soc. Lecture Note Ser.
| title = Surveys in combinatorics, 1997 (London)
| volume = 241
| year = 1997}}.</ref> This duality extends to nonplanar graphs as well, via the [[Tutte polynomial]] <math>T_G</math> of a graph <math>G</math>, which can be used to count both types of orientations: the number of acyclic orientations of <math>G</math> is <math>T_G(2,0)</math> and the number of totally cyclic orientations is <math>T_G(0,2)</math>.<ref>{{citation
| last = Las Vergnas | first = Michel | authorlink = Michel Las Vergnas
| doi = 10.1016/0095-8956(80)90082-9
| issue = 2
| journal = [[Journal of Combinatorial Theory]] | series = Series B
| mr = 586435
| pages = 231–243
| title = Convexity in oriented matroids
| volume = 29
| year = 1980}}.</ref> The number of acyclic orientations may also be counted using a different polynomial, the [[chromatic polynomial]] <math>\chi_G</math>: there are <math>|\chi_G(-1)|</math> different acyclic orientations.<ref>{{citation|first=Richard P.|last=Stanley|authorlink=Richard P. Stanley|title=Acyclic orientations of graphs|journal=Discrete Mathematics|volume=5|issue=2 |pages=171–178|year= 1973|doi=10.1016/0012-365X(73)90108-8}}.</ref>
The set of all acyclic orientations of a given graph may be given the structure of a [[partial cube]], in which two acyclic orientations are adjacent whenever they differ in the direction of a single edge.<ref>{{citation
| last1 = Fukuda | first1 = Komei
| last2 = Prodon | first2 = Alain
| last3 = Sakuma | first3 = Tadashi
| doi = 10.1016/S0304-3975(00)00226-7
| issue = 1-2
| journal = Theoretical Computer Science
| mr = 1846912
| pages = 9–16
| title = Notes on acyclic orientations and the shelling lemma
| url = ftp://ftp.ifor.math.ethz.ch/pub/fukuda/paper/acyclic980112.ps.gz
| volume = 263
| year = 2001}}.</ref>
 
An acyclic orientation of a [[complete graph]] is called a [[Tournament (graph theory)|transitive tournament]], and is equivalent to a [[total ordering]] of the graph's vertices. In such an orientation there is in particular exactly one source and exactly one sink; more generally, an acyclic orientation with a unique source and a unique sink is called a [[bipolar orientation]].<ref>{{citation
| last1 = de Fraysseix | first1 = Hubert
| last2 = de Mendez | first2 = Patrice Ossona
| last3 = Rosenstiehl | first3 = Pierre | author3-link = Rosenstiehl
| doi = 10.1016/0166-218X(94)00085-R
| issue = 2-3
| journal = Discrete Applied Mathematics
| mr = 1318743
| pages = 157–179
| title = Bipolar orientations revisited
| volume = 56
| year = 1995}}.</ref>
 
==References==
{{reflist}}
 
[[Category:Graph theory objects]]

Latest revision as of 22:46, 19 September 2014

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