FKT algorithm: Difference between revisions

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The '''Erdős–Gallai theorem''' is a result in [[graph theory]], a branch of [[combinatorics]], that gives a necessary and sufficient condition for a [[finite sequence]] of [[natural number]]s to be the [[Degree (graph theory)#Degree sequence|degree sequence]] of a [[simple graph]]; a sequence obeying these conditions is called "graphic". The theorem was published in 1960 by [[Paul Erdős]] and [[Tibor Gallai]], after whom it is named.
The London schedule has long had cheerleaders for its talent on home territory, but in the past it has been squeezed out by more commercially robust cities.<br><br>Now it seems the capital is holding its ground against the bigger boys, as more and more international press, and buyers, make London an important stop in their schedule - blizzards in New York notwithstanding.<br>This in part has been due to brands like Burberry Prorsum, Mulberry and Tom Ford, which, with the weight of their advertising budgets behind them, command an international audience. Topshop, led by creative director Kate Phelan, also of British Vogue, has also helped provide financial support and a professional framework in which the creativity of London�s emerging designers can be showcased.<br><br>This situation has not gone unnoticed in Milan, where Giorgio Armani has taken Stella Jean under his wing, hosting her spring/summer 2014 show last September. �The new generation of Italian designers needs our support,� said Armani, echoing the sentiment that has long made London great<br>
LVMH (the parent company of brands such as Louis Vuitton, Christian Dior and [http://www.pcs-systems.co.uk/Images/celinebag.aspx Cheap Celine Bags]) is also taking lessons from London: on Thursday it announced a shortlist of 30 designers for its inaugural LVMH Young Fashion Designer Prize. With �300,000 (�246,000) at stake, plus a year of mentoring, the number of London designers shortlisted, including Simone Rocha, Thomas Tait, Meadham Kirchhoff and J JS Lee, is a clear indication of the capital�s fertile groun<br><br>
Fittingly, the nominees - many of which have come through support programmes Newgen and Fashion East - are some of the most exciting shows on the schedul<br>
Korean-born J JS Lee, who trained at Central Saint Martins, took the prized opening slot yesterday in rain-swept central London. For autumn/winter 2014 she showed an elegant collection of brushed mohair, plaid overcoats and formal dresses - experimenting with cocoon shapes, biker jackets and rounded shoulders for the former, while playing with different lengths for the latte<br><br>
Cobalt blue houndstooth as a laser-cut detail and d�grad� printed silk provided a jolt of colour in a predominantly cream, navy and forest-green palette. Slim, long-line trousers were an elegant proposition for evenings, while chunky, cable knits were covetably co<br>.
The London College of Fashion MA Show at the Waldorf Astoria Hotel on day one of London Fashion Week yesterday (PA) With a fit-to-burst schedule of shows, presentations and parties over the next four days, there�s plenty to inspire and excite the fashion world - both on and off the catwa<br><br>


==Theorem statement==
JW Anderson�s first womenswear collection since LVMH bought a minority stake in his business last September is sure to be as controversial as ever: the designer�s gender-blurring take on dressing has its detractors, but just as many fans. The designer was also appointed creative director of Loewe, a Spanish leather brand within the stable, so an excellent accessories offering is to be expect<br><br>
A sequence of non-negative [[integer]]s <math>d_1\geq\cdots\geq d_n</math> can be represented as the degree sequence of a finite [[simple graph]] on ''n'' vertices if and only if <math>d_1+\cdots+d_n</math> is even and


: <math>
It was all change at Mulberry last season too, as [http://En.Search.Wordpress.com/?q=Emma+Hill Emma Hill] departed after five years as creative director of the British brand. Hill seemed to have hit on a winning formula: balancing design and the right kind of celebrity endorsements fortified the brand�s finances, but recent profit warnings are cited as the reason for her departu<br><br>
\sum^{k}_{i=1}d_i\leq k(k-1)+ \sum^n_{i=k+1} \min(d_i,k)
</math>


holds for <math>1\leq k\leq n</math>.
This season, the brand will not be hosting its usual catwalk extravaganza in Claridge�s ballroom. Instead it has announced a photocall at which Cara Delevingne, its campaign face, will reveal the brand�s next big move. Speculation is rife that she will be announced as creative director, but the brand�s success with bags named after It girls makes the debut of the �Cara� a far more likely prosp<br><br>


==Proofs==
Joseph, celebrating 25 years since its first west London boutique opened, is presenting on schedule for the first time too. It may not have the budgets of its blockbuster digital rivals, but it is held in esteem by serious shoppers and the fashion cr<br><br><br>
It is not difficult to show that the conditions of the Erdős–Gallai theorem are necessary for a sequence of numbers to be graphic. The requirement that the sum of the degrees be even is the [[handshaking lemma]], already used by Euler in his 1736 paper on the [[bridges of Königsberg]]. The inequality between the sum of the <math>k</math> largest degrees and the sum of the remaining degrees can be established by [[double counting (proof technique)|double counting]]: the left side gives the numbers of edge-vertex adjacencies among the <math>k</math> highest-degree vertices, each such adjacency must either be on an edge with one or two high-degree endpoints, the <math>k(k-1)</math> term on the right gives the maximum possible number of edge-vertex adjacencies in which both endpoints have high degree, and the remaining term on the right upper bounds the number of edges that have exactly one high degree endpoint. Thus, the more difficult part of the proof is to show that, for any sequence of numbers obeying these conditions, there exists a graph for which it is the degree sequence.


The original proof of {{harvtxt|Erdős|Gallai|1960}} was long and involved. {{harvtxt|Choudom|1986}} cites a shorter proof by [[Claude Berge]], based on ideas of [[flow network|network flow]]. Choudom instead provides a proof by [[mathematical induction]] on the sum of the degrees: he lets <math>t</math> be the first index of a number in the sequence for which <math>d_t > d_{t+1}</math> (or the penultimate number if all are equal), uses a case analysis to show that the sequence formed by subtracting one from <math>d_t</math> and from the last number in the sequence (and removing the last number if this subtraction causes it to become zero) is again graphic, and forms a graph representing the original sequence by adding an edge between the two positions from which one was subtracted.
While there is a respectable coterie of British luxury brands, it�s the commercial pull of the high street that is unrivalled, something the British Fashion Council - responsible for co-ordinating the schedule - has cottoned on to. As well as Topshop�s premium offering Unique, now a regular on the schedule, the BFC has courted brands such as River Island, which debuted its collaboration with Rihanna in London this time last y<br>r.
Now Whistles is presenting on the schedule for autumn/winter 2014; with a mens collection rumoured to be on the horizon and a flagship store on Mayfair�s Dover Street, the brand is certainly looking to exp<br><br>


{{harvtxt|Tripathi|Venugopalan|West|2010}} consider a sequence of "subrealizations", graphs whose degrees are upper bounded by the given degree sequence. They show that, if ''G'' is a subrealization, and ''i'' is the smallest index of a vertex in ''G'' whose degree is not equal to ''d<sub>i</sub>'', then ''G'' may be modified in a way that produces another subrealization, increasing the degree of vertex ''i'' without changing the degrees of the earlier vertices in the sequence. Repeated steps of this kind must eventually reach a realization of the given sequence, proving the theorem.
So too are Christopher Kane and Roksanda Ilincic, who plan to open boutiques on Mount Street this year.
 
==Relation to integer partitions==
{{harvtxt|Aigner|Triesch|1994}} describe close connections between the Erdős–Gallai theorem and the theory of [[Partition (number theory)|integer partitions]].
Let <math>m=\sum d_i</math>; then the sorted integer sequences summing to <math>m</math> may be interpreted as the partitions of <math>m</math>. Under [[majorization]] of their [[prefix sum]]s, the partitions form a [[Lattice (order)|lattice]], in which the minimal change between an individual partition and another partition lower in the partition order is to subtract one from one of the numbers <math>d_i</math> and add it to a number <math>d_{i+1}</math> that is smaller by at least two. As Aigner and Triesch show, this operation preserves the property of being graphic, so to prove the Erdős–Gallai theorem it suffices to characterize the graphic sequences that are maximal in this majorization order. They provide such a characterization, in terms of the [[Ferrers diagram]]s of the corresponding partitions, and show that it is equivalent to the Erdős–Gallai theorem.
 
==References==
*{{citation
| last1 = Aigner | first1 = Martin
| last2 = Triesch | first2 = Eberhard
| doi = 10.1016/0012-365X(94)00104-Q
| issue = 1-3
| journal = Discrete Mathematics
| mr = 1313278
| pages = 3–20
| title = Realizability and uniqueness in graphs
| volume = 136
| year = 1994}}.
*{{citation
| last = Choudum | first = S. A. | author-link = S. A. Choudum
| doi = 10.1017/S0004972700002872
| issue = 1
| journal = Bulletin of the Australian Mathematical Society
| mr = 823853
| pages = 67–70
| title = A simple proof of the Erdős-Gallai theorem on graph sequences
| volume = 33
| year = 1986}}.
*{{citation
| last1 = Erdős | first1 = P. | author1-link = Paul Erdős
| last2 = Gallai | first2 = T. | author2-link = Tibor Gallai
| journal = Matematikai Lapok
| pages = 264–274
| title = Gráfok előírt fokszámú pontokkal
| url = http://www.renyi.hu/~p_erdos/1961-05.pdf
| volume = 11
| year = 1960}}.
*{{citation
| last1 = Tripathi | first1 = Amitabha
| last2 = Venugopalan | first2 = Sushmita
| last3 = West | first3 = Douglas B. | author3-link = Douglas West (mathematician)
| doi = 10.1016/j.disc.2009.09.023
| issue = 4
| journal = Discrete Mathematics
| mr = 2574834
| pages = 843–844
| title = A short constructive proof of the Erdős-Gallai characterization of graphic lists
| url = http://www.math.uiuc.edu/~west/pubs/tripathi.ps
| volume = 310
| year = 2010}}
 
{{DEFAULTSORT:Erdos-Gallai theorem}}
[[Category:Paul Erdős]]
[[Category:Theorems in graph theory]]

Latest revision as of 20:38, 19 May 2014

The London schedule has long had cheerleaders for its talent on home territory, but in the past it has been squeezed out by more commercially robust cities.

Now it seems the capital is holding its ground against the bigger boys, as more and more international press, and buyers, make London an important stop in their schedule - blizzards in New York notwithstanding.
This in part has been due to brands like Burberry Prorsum, Mulberry and Tom Ford, which, with the weight of their advertising budgets behind them, command an international audience. Topshop, led by creative director Kate Phelan, also of British Vogue, has also helped provide financial support and a professional framework in which the creativity of London�s emerging designers can be showcased.

This situation has not gone unnoticed in Milan, where Giorgio Armani has taken Stella Jean under his wing, hosting her spring/summer 2014 show last September. �The new generation of Italian designers needs our support,� said Armani, echoing the sentiment that has long made London great
LVMH (the parent company of brands such as Louis Vuitton, Christian Dior and Cheap Celine Bags) is also taking lessons from London: on Thursday it announced a shortlist of 30 designers for its inaugural LVMH Young Fashion Designer Prize. With �300,000 (�246,000) at stake, plus a year of mentoring, the number of London designers shortlisted, including Simone Rocha, Thomas Tait, Meadham Kirchhoff and J JS Lee, is a clear indication of the capital�s fertile groun

Fittingly, the nominees - many of which have come through support programmes Newgen and Fashion East - are some of the most exciting shows on the schedul
Korean-born J JS Lee, who trained at Central Saint Martins, took the prized opening slot yesterday in rain-swept central London. For autumn/winter 2014 she showed an elegant collection of brushed mohair, plaid overcoats and formal dresses - experimenting with cocoon shapes, biker jackets and rounded shoulders for the former, while playing with different lengths for the latte

Cobalt blue houndstooth as a laser-cut detail and d�grad� printed silk provided a jolt of colour in a predominantly cream, navy and forest-green palette. Slim, long-line trousers were an elegant proposition for evenings, while chunky, cable knits were covetably co
. The London College of Fashion MA Show at the Waldorf Astoria Hotel on day one of London Fashion Week yesterday (PA) With a fit-to-burst schedule of shows, presentations and parties over the next four days, there�s plenty to inspire and excite the fashion world - both on and off the catwa

JW Anderson�s first womenswear collection since LVMH bought a minority stake in his business last September is sure to be as controversial as ever: the designer�s gender-blurring take on dressing has its detractors, but just as many fans. The designer was also appointed creative director of Loewe, a Spanish leather brand within the stable, so an excellent accessories offering is to be expect

It was all change at Mulberry last season too, as Emma Hill departed after five years as creative director of the British brand. Hill seemed to have hit on a winning formula: balancing design and the right kind of celebrity endorsements fortified the brand�s finances, but recent profit warnings are cited as the reason for her departu

This season, the brand will not be hosting its usual catwalk extravaganza in Claridge�s ballroom. Instead it has announced a photocall at which Cara Delevingne, its campaign face, will reveal the brand�s next big move. Speculation is rife that she will be announced as creative director, but the brand�s success with bags named after It girls makes the debut of the �Cara� a far more likely prosp

Joseph, celebrating 25 years since its first west London boutique opened, is presenting on schedule for the first time too. It may not have the budgets of its blockbuster digital rivals, but it is held in esteem by serious shoppers and the fashion cr


While there is a respectable coterie of British luxury brands, it�s the commercial pull of the high street that is unrivalled, something the British Fashion Council - responsible for co-ordinating the schedule - has cottoned on to. As well as Topshop�s premium offering Unique, now a regular on the schedule, the BFC has courted brands such as River Island, which debuted its collaboration with Rihanna in London this time last y
r. Now Whistles is presenting on the schedule for autumn/winter 2014; with a mens collection rumoured to be on the horizon and a flagship store on Mayfair�s Dover Street, the brand is certainly looking to exp

So too are Christopher Kane and Roksanda Ilincic, who plan to open boutiques on Mount Street this year.

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