en>Wcherowi: /* Construction via embeddings */ removed DAB

2012-12-05T02:34:33Z

Construction via embeddings: removed DAB

← Older revision		Revision as of 04:34, 5 December 2012
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	~~Hello~~ and ~~welcome~~. ~~My name~~ is ~~Irwin~~ and ~~I totally dig~~ that ~~name~~. ~~To do aerobics~~ is a ~~thing~~ that I'~~m totally addicted~~ to. ~~In her expert life she~~ is a ~~payroll clerk~~ but ~~she~~'s ~~always needed her own company~~. ~~South Dakota is exactly where me~~ and ~~my spouse live~~ and ~~my family loves it~~.<br><br>~~My blog;~~ [http://www.~~breda~~.nl/~~users~~/~~noeliadfebdftijfsdnt home std test kit~~]		In [[graph theory]], '''nowhere-zero flows''' are a special type of [[Flow network\|network flow]] which is related (by duality) to [[graph coloring\|coloring]] [[planar graphs]].

			== Definition ==
			Let ''G'' = ''(V,E)'' be a [[directed graph]] and let ''M'' be an [[abelian group]]. A map φ: ''E'' → ''M'' is a '''flow''' or an ''M''-'''flow''' if for every vertex ''v'' ∈ ''V'', it holds that
			:<math>\sum_{e \in \delta^+(v)} \phi(e) = \sum_{e \in \delta^-(v)} \phi(e),</math>
			where ''δ<sup>+</sup>(v)'' denotes the set of edges out of ''v'' and ''δ<sup>–</sup>(v)'' denotes the set of edges into ''v''.
			Sometimes, this condition is referred to as [[Kirchhoff's circuit laws\|Kirchhoff's law]].
			If ''φ(e)'' ≠ 0 for every ''e'' ∈ ''E'', we call φ a '''nowhere-zero''' flow. If ''M'' = '''Z''' is the group of integers under addition and ''k'' is a positive integer with the property that –''k'' < ''φ(e)'' < ''k'' for every edge ''e'', then the ''M''-flow φ is also called a ''k''-'''flow'''.

			Let ''G'' = ''(V,E)'' be an undirected graph. An orientation of ''E'' is a '''modular''' ''k''-'''flow''' if
			:<math>\|\delta^+(v)\| \equiv \|\delta^-(v)\| \pmod{k}</math>
			for every vertex ''v'' ∈ ''V''.

			== Properties ==

			Modify a nowhere-zero flow φ on a graph ''G'' by choosing an edge ''e'', reversing it, and then replacing ''φ(e)'' with ''-φ(e)''. After this adjustment, φ is still a nowhere-zero flow. Furthermore, if φ was originally a ''k''-flow, then the resulting φ is also a ''k''-flow. Thus, the existence of a nowhere-zero ''M''-flow or a nowhere-zero ''k''-flow is independent of the orientation of the graph. Thus, an undirected graph ''G'' is said to have a nowhere-zero ''M''-flow or nowhere-zero ''k''-flow if some (and thus every) orientation of ''G'' has such a flow.

			More surprisingly, if ''M'' is a finite abelian group of size ''k'', then the number of a nowhere-zero ''M''-flows in some graph does not depend on the structure of ''M'' but only on ''k'', the size of ''M''. Furthermore, the existence of a ''M''-flow coincides with the existence of a ''k''-flow. These two results were proved by [[Tutte]] in 1953.<ref>{{cite journal
			\| last = Tutte
			\| first = William Thomas
			\| authorlink = W. T. Tutte
			\| year = 1953
			\| title = A contribution to the theory of chromatic polynomials
			\| url = http://cms.math.ca/cjm/a144778#
			}}</ref>

			== Flow/coloring duality ==
			Let ''G'' = ''(V,E)'' be a directed [[Bridge (graph theory)\|bridgeless]] graph drawn in the plane, and assume that the regions of this drawing are properly ''k''-colored with the colors {0, 1, 2, .., ''k'' – 1}. Now, construct a map φ: ''E(G)'' → {–(''k'' – 1), ..., –1, 0, 1, ..., ''k'' – 1} by the following rule: if the edge ''e'' has a region of color ''x'' to the left and a region of color ''y'' to the right, then let ''φ(e)'' = ''x'' – ''y''. It is an easy exercise to show that φ is a ''k''-flow. Furthermore, since the regions were properly colored, φ is a nowhere-zero ''k''-flow. It follows from this construction, that if ''G'' and ''G'' are planar dual graphs and ''G'' is ''k''-colorable, then ''G'' has a nowhere-zero ''k''-flow. Tutte proved that the converse of this statement is also true. Thus, for planar graphs, nowhere-zero flows are dual to colorings. Since nowhere-zero flows make sense for general graphs (not just graphs drawn in the plane), this study can be viewed as an extension of coloring theory for non-planar graphs.

			== Theory ==
			{{unsolved\|mathematics\|Does every bridgeless graph have a nowhere zero 5-flow? Does every bridgeless graph that does not have the Petersen graph as a minor have a nowhere zero 4-flow?}}
			Just as no graph with a [[glossary of graph theory\|loop]] edge has a proper coloring, no graph with a [[glossary of graph theory\|bridge]] can have a nowhere-zero flow (in any group). It is easy to show that every graph without a bridge has a nowhere-zero '''Z'''-flow (a form of [[Robbins theorem]]), but interesting questions arise when trying to find nowhere-zero ''k''-flows for small values of ''k''. Two nice theorems in this direction are Jaeger's 4-flow theorem (every 4-[[glossary of graph theory\|edge-connected]] graph has a nowhere-zero 4-flow)<ref>F. Jaeger, Flows and generalized coloring theorems in graphs, J. Comb. Theory Set. B, 26 (1979), 205-216.</ref> and Seymour's 6-flow theorem (every bridgeless graph has a nowhere-zero 6-flow).<ref>P. D. Seymour, Nowhere-zero 6-flows, J. Comb. Theory Ser B, 30 (1981), 130-135.</ref>

			[[W. T. Tutte\|Tutte]] conjectured that every bridgeless graph has a nowhere-zero 5-flow<ref>[http://garden.irmacs.sfu.ca/?q=op/5_flow_conjecture 5-flow conjecture], Open Problem Garden.</ref> and that every bridgeless graph that does not have the [[Petersen graph]] as a [[minor (graph theory)\|minor]] has a nowhere-zero 4-flow.<ref>[http://garden.irmacs.sfu.ca/?q=op/4_flow_conjecture 4-flow conjecture], Open Problem Garden.</ref> For [[cubic graph]]s with no Petersen minor, a 4-flow is known to exist as a consequence of the [[Snark (graph theory)\|snark theorem]] but for arbitrary graphs these conjectures remain open.

			== See also ==
			* [[Cycle space]]

			==References==
			{{reflist}}
			* {{cite book
			\| last = Zhang
			\| first = Cun-Quan
			\| title = Integer Flows and Cycle Covers of Graphs
			\| url = http://www.math.wvu.edu/~cqzhang/book.html
			\| series = Chapman & Hall/CRC Pure and Applied Mathematics Series
			\| year = 1997
			\| publisher = Marcel Dekker, Inc.
			\| isbn = 9780824797904
			\| lccn = 96037152
			}}
			* T.R. Jensen and B. Toft, Graph Coloring Problems, Wiley-Interscience Serires in Discrete Mathematics and Optimization, (1995)

			[[Category:Network flow]]

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2011-02-19T12:57:01Z

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Hello and welcome. My name is Irwin and I totally dig that name. To do aerobics is a thing that I'm totally addicted to. In her expert life she is a payroll clerk but she's always needed her own company. South Dakota is exactly where me and my spouse live and my family loves it.<br><br>My blog; [http://www.breda.nl/users/noeliadfebdftijfsdnt home std test kit]

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en>Wcherowi: /* Construction via embeddings */ removed DAB

en>SmackBot: Dated {{Technical}}. (Build p607)