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Chen, Boliong; Matsumoto, Makoto; Wang, Jianfang; Zhang, Zhongfu; Zhang, Jianxun (1994-03-01). "A short proof of Nash-Williams' theorem for the arboricity of a graph".
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The Nash-Williams partition theorem (Formal proof development in
Isabelle/HOL, Archive of Formal Proofs)
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that saturates the inequality (or else we can choose a smaller t). This leads to the following formula
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Theorem in graph theory describing number of edge-disjoint spanning trees a graph can have
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Nash-Williams, Crispin St. John Alvah. "Decomposition of Finite Graphs Into
Forests".
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Nash-Williams, Crispin St. John Alvah. "Decomposition of Finite Graphs Into
Forests".
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The general problem is to ask when a graph can be covered by edge-disjoint subgraphs.
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507:{\displaystyle t=\lceil \max _{S\subset G}{\frac {E(S)}{V(S)-1}}\rceil }
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This is how people usually define what it means for a graph to be
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is slightly different and applies to forests rather than trees.)
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In 1964, Nash-Williams generalized the above result to forests:
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edge-disjoint spanning trees iff for every partition
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For this article, we will say that such a graph has
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33:theorem that describes how many edge-disjoint
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107:{\textstyle V_{1},\ldots ,V_{k}\subset V(G)}
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424:. It is tight in that there is a subgraph
227:edge-disjoint paths between two vertices.
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346:In other words, for every subgraph
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231:Nash-Williams theorem for forests
600:Diestel, Reinhard (2017-06-30).
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190:Related tree-packing properties
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271:{\displaystyle U\subset V(G)}
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182:. (The actual definition of
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662:Graphs and Combinatorics
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517:also referred to as the
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552:Tree packing conjecture
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37:(and more generally
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519:NW formula.
620:1048203362
557:References
533:Arboricity
354:, we have
216:-arboric.
184:arboricity
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682:1435-5914
641:(1): 12.
502:⌉
493:−
456:⊂
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365:≥
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135:∅
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719:Category
527:See also
342:-aboric.
45:A graph
350:=
330:edges.
180:arboric
39:forests
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