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| The '''arboricity''' of an [[undirected graph]] is the minimum number of [[Tree (graph theory)|forests]] into which its edges can be [[graph partition|partitioned]]. Equivalently it is the minimum number of [[spanning tree (mathematics)|spanning forests]] needed to cover all the edges of the graph.
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| ==Example==
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| [[Image:K44 arboricity.svg|thumb|A partition of the [[complete bipartite graph]] ''K''<sub>4,4</sub> into three forests, showing that it has arboricity three.]]
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| The figure shows the [[complete bipartite graph]] ''K''<sub>4,4</sub>, with the colors indicating a partition of its edges into three forests. ''K''<sub>4,4</sub> cannot be partitioned into fewer forests, because any forest on its eight vertices has at most seven edges, while the overall graph has sixteen edges, more than double the number of edges in a single forest. Therefore, the arboricity of ''K''<sub>4,4</sub> is three.
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| ==Arboricity as a measure of density==
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| The arboricity of a graph is a measure of how [[dense graph|dense]] the graph is: graphs with many edges have high arboricity, and graphs with high arboricity must have a dense subgraph.
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| In more detail, as any n-vertex forest has at most n-1 edges, the arboricity of a graph with n vertices and m edges is at least <math>\lceil m/(n-1)\rceil</math>. Additionally, the subgraphs of any graph cannot have arboricity larger than the graph itself, or equivalently the arboricity of a graph must be at least the maximum arboricity of any of its subgraphs. [[Crispin St. J. A. Nash-Williams|Nash-Williams]] proved that these two facts can be combined to characterize arboricity: if we let n<sub>S</sub> and m<sub>S</sub> denote the number of vertices and edges, respectively, of any subgraph S of the given graph, then the arboricity of the graph equals <math>\max\{\lceil m_S/(n_S-1)\rceil\}.</math>
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| Any [[planar graph]] with <math>n</math> vertices has at most <math>3n-6</math> edges, from which it follows by Nash-Williams' formula that planar graphs have arboricity at most three. Schnyder used a special decomposition of a planar graph into three forests called a '''Schnyder wood''' to find a [[Fáry's theorem|straight-line embedding]] of any planar graph into a grid of small area.
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| ==Algorithms==
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| The arboricity of a graph can be expressed as a special case of a more general [[matroid partitioning]] problem, in which one wishes to express a set of elements of a matroid as a union of a small number of independent sets. As a consequence, the arboricity can be calculated by a polynomial-time algorithm {{Harv|Gabow|Westermann|1992}}.
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| == Related concepts ==
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| The '''star arboricity''' of a graph is the size of the minimum forest, each tree of which is a [[Star (graph theory)|star]] (tree with at most one non-leaf node), into which the edges of the graph can be partitioned. If a tree is not a star itself, its star arboricity is two, as can be seen by partitioning the edges into two subsets at odd and even distances from the tree root respectively. Therefore, the star arboricity of any graph is at least equal to the arboricity, and at most equal to twice the arboricity.
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| The '''linear arboricity''' of a graph is the size of the minimum '''linear forest''' (a [[Forest (graph theory)|forest]] in which all vertices are incident to at most two edges) into which the edges of the graph can be partitioned. The linear arboricity of a graph is closely related to its maximum [[Degree (graph theory)|degree]] and its [[slope number]].
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| The '''[[pseudoarboricity]]''' of a graph is the minimum number of [[pseudoforest]]s into which its edges can be partitioned. Equivalently, it is the maximum ratio of edges to vertices in any subgraph of the graph. As with the arboricity, the pseudoarboricity has a matroid structure allowing it to be computed efficiently {{Harv|Gabow|Westermann|1992}}.
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| The '''[[Glossary of graph theory#Genus|thickness]]''' of a graph is the minimum number of planar subgraphs into which its edges can be partitioned. As any planar graph has arboricity three, the thickness of any graph is at least equal to a third of the arboricity, and at most equal to the arboricity.
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| The '''[[Degeneracy (graph theory)|degeneracy]]''' of a graph is the maximum, over all [[induced subgraph]]s of the graph, of the minimum [[Degree (graph theory)|degree]] of a vertex in the subgraph. The degeneracy of a graph with arboricity <math>a</math> is at least equal to <math>a</math>, and at most equal to <math>2a-1</math>. The coloring number of a graph, also known as its Szekeres-Wilf number {{Harv|Szekeres|Wilf|1968}} is always equal to its degeneracy plus 1 {{Harv|Jensen|Toft|1995|p=77f.}}.
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| The '''[[Strength_of_a_graph|strength]]''' of a graph is a fractional value whose integer part gives the maximum number of disjoint spanning trees that can be drawn in a graph. It is the packing problem that is dual to the covering problem raised by the arboricity. The two parameters have been studied together by Tutte and Nash-Williams.
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| ==Notes==
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| {{reflist}}
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| == References ==
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| {{refbegin}}
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| * {{Cite journal|authorlink=Noga Alon|first=N.|last=Alon|title=The linear arboricity of graphs|journal=[[Israel Journal of Mathematics]]|volume=62|issue=3|year=1988|pages=311–325|doi=10.1007/BF02783300|mr=0955135|ref=harv}}
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| *{{cite journal|first1=B.|last1=Chen|first2=M.|last2=Matsumoto|first3=J.|last3=Wang|first4=Z.|last4=Zhang|first5= J.|last5=Zhang|title=A short proof of Nash-Williams' theorem for the arboricity of a graph|journal=Graphs and Combinatorics|volume=10|issue=1|year=1994|pages=27–28|doi=10.1007/BF01202467|mr=1273008|ref=harv}}
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| *{{Cite journal|first1=P.|last1=Erdős|author1-link=Paul Erdős|first2=A.|last2=Hajnal|author2-link=András Hajnal|title=On chromatic number of graphs and set-systems|journal=[[Acta Mathematica Hungarica]]|volume=17|issue=1–2|pages=61–99|year=1966|url=http://www.math-inst.hu/~p_erdos/1966-07.pdf|doi=10.1007/BF02020444|mr=0193025|ref=harv}}
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| *{{Cite journal|first1=H. N.|last1=Gabow|first2=H. H.|last2=Westermann|title=Forests, frames, and games: Algorithms for matroid sums and applications|journal=[[Algorithmica]]|volume=7|issue=1|year=1992|pages=465–497|mr=1154585|doi=10.1007/BF01758774|ref=harv}}
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| *{{cite journal|first1=S. L.|last1=Hakimi|author1-link=S. L. Hakimi |first2=J.|last2=Mitchem|first3=E. E.|last3=Schmeichel|title=Star arboricity of graphs|journal=[[Discrete Mathematics (journal)|Discrete Mathematics]]|volume=149|year=1996|pages=93–98|mr=1375101|doi=10.1016/0012-365X(94)00313-8|ref=harv}}
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| *{{Cite book
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| | last1 = Jensen | first1 = T. R.
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| | last2 = Toft | first2 = B.
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| | title = Graph Coloring Problems
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| | publisher = Wiley-Interscience
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| | location = New York
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| | year = 1995
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| | isbn = 0-471-02865-7
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| | mr = 1304254
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| | ref = harv}}
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| *{{cite journal|author=C. St. J. A. Nash-Williams|authorlink=Crispin St. J. A. Nash-Williams|title=Edge-disjoint spanning trees of finite graphs|journal=[[Journal of the London Mathematical Society]]|volume=36|issue=1|year=1961|pages=445–450|doi=10.1112/jlms/s1-36.1.445|mr=0133253|ref=harv}}
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| *{{cite journal|author=C. St. J. A. Nash-Williams|authorlink=Crispin St. J. A. Nash-Williams|title=Decomposition of finite graphs into forests|journal=[[Journal of the London Mathematical Society]]|volume=39|issue=1|year=1964|pages=12|doi=10.1112/jlms/s1-39.1.12|mr=0161333|ref=harv}}
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| *{{cite conference|author=W. Schnyder|url=http://portal.acm.org/citation.cfm?id=320176.320191|title=Embedding planar graphs on the grid|booktitle=Proc. 1st ACM/SIAM Symposium on Discrete Algorithms (SODA)|year=1990|pages=138–148}}
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| *{{Cite journal|first1=G.|last1=Szekeres|author1-link=George Szekeres|first2=H. S.|last2=Wilf|author2-link=Herbert Wilf|title=An inequality for the chromatic number of a graph|journal=[[Journal of Combinatorial Theory]]|year=1968|ref=harv|mr=0218269}}
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| *{{cite journal|first=W. T.|last=Tutte|authorlink=W. T. Tutte|title=On the problem of decomposing a graph into n connected factors|journal=[[Journal of the London Mathematical Society]]|volume=36|issue=1|year=1961|pages=221–230|doi=10.1112/jlms/s1-36.1.221|mr=0140438|ref=harv}}
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| {{refend}}
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| [[Category:Graph invariants]]
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| [[Category:Spanning tree]]
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