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In the [[mathematical]] discipline of [[graph theory]], a '''graph labeling''' is the assignment of labels, traditionally represented by [[integers]], to the [[edge (graph theory)|edges]] or [[vertex (graph theory)|vertices]], or both, of a [[Graph (mathematics)|graph]].<ref name=mathw>{{mathworld|LabeledGraph|Labeled  graph}}</ref>
 
Formally, given a graph ''G'', a '''vertex labeling''' is a function  of ''G'' to a set of ''labels''. A graph with such a function defined is called a '''vertex-labeled graph'''. Likewise, an '''edge labeling''' is a function mapping edges of ''G'' to a set of "labels". In this case, ''G'' is called an '''edge-labeled graph'''.
When the edge labels are members of an ordered set (e.g., the [[real number]]s), it may be called a '''weighted graph'''.
 
When used without qualification, the term '''labeled graph''' generally refers to a vertex-labeled graph with all labels distinct. Such a graph may equivalently be labeled by the consecutive integers {1,&nbsp;...,&nbsp;''n''}, where ''n'' is the number of vertices in the graph.<ref name=mathw/>
For many applications, the edges or vertices are given labels that are meaningful in the associated domain. For example, the edges may be assigned [[Weighted graph|weights]] representing the "cost" of traversing between the incident vertices.<ref>"Different Aspects of Coding Theory", by Robert Calderbank (1995) ISBN 0-8218-0379-4, [http://books.google.com/books?id=TcOzdq3nDp4C&pg=PA57&dq=%22labeled+graph%22&lr=#PPA53,M1 p. 53]"</ref>
 
In the above definition a graph is understood to be a finite undirected simple graph. However, the notion of labeling may be applied to all extensions and generalizations of graphs. For example, in [[automata theory]] and [[formal language]] theory it is convenient to consider labeled [[multigraph]]s, i.e., a pair of vertices may be connected by several labeled edges.<ref>"[[Developments in Language Theory]]", Proc. 9th. Internat.Conf., 2005, ISBN 3-540-26546-5, [http://books.google.com/books?id=QPgojKbuuUEC&pg=PA314&dq=%22labeled+graph%22#PPA313,M1 p. 313] </ref>
 
==History==
Most graph labelings trace their origins to labelings presented by Alex Rosa in his 1967 paper.<ref name="Gallian">{{cite journal|author=Gallian, J.|title=A Dynamic Survey of Graph Labelings, 1996-2005|publisher=The Electronic Journal of Combinatorics}}</ref> Rosa identified three types of labelings, which he called α-, β-, and ρ-labelings.<ref name="Rosa">{{cite journal|author=Rosa, A.|title=On certain valuations of vertices in a graph}}</ref> β-Labelings were later renamed '''graceful''' by [[Solomon Golomb|S.W. Golomb]] and the name has been popular since.
 
==Special cases==
===Graceful labeling===
{{main|Graceful labeling}}
[[File:Graceful labeling.svg|thumb|A graceful labeling. Vertex labels are in black, edge labels in red]]
 
A graph is known as graceful when its vertices are labeled from 0 to <math>\|E\|</math>, the size of the graph, and this labeling induces an edge labeling from 1 to <math>\|E\|</math>. For any edge ''e'', ''e'''s label is the positive difference between the two vertices incident with e. In other words, if ''e'' is incident with vertices labeled ''k'' and ''j'', ''e'' will be labeled <math>\|k - j\|</math>. Thus, a graph <math>G:=(V,E)</math> is graceful if and only if there exists an injection that induces a bijection from ''E'' to the positive integers up to <math>\|E\|</math>.
 
In his original paper, Rosa proved that all [[eulerian graph]]s with order [[Equivalence relation|equivalent]] to 1 or 2 ([[Modular arithmetic|mod]] 4) are not graceful. Whether or not certain families of graphs are graceful is an area of graph theory under extensive study. Arguably, the largest unproven conjecture in Graph Labeling is the Ringel-Kotzig conjecture, which hypothesizes that all trees are graceful. This has been proven for all [[path graph|paths]], [[caterpillar tree|caterpillars]], and many other infinite families of trees. Kotzig himself has called the effort to prove the conjecture a "disease."<ref>{{cite journal | zbl=1163.05007 | last=Vietri | first=Andrea | title=Sailing towards, and then against, the graceful tree conjecture: some promiscuous results | journal=Bull. Inst. Comb. Appl. | volume=53 | pages=31–46 | year=2008 | issn=1183-1278 }}</ref>
 
===Edge-graceful labeling===
{{main|Edge-graceful labeling}}
An '''edge-graceful''' labeling on a simple graph (no loops or multiple edges) on ''p'' vertices and ''q'' edges is a labelling of the edges by distinct integers in {1,...,''q''} such that the labeling on the vertices induced by labeling a vertex with the sum of the incident edges taken modulo ''p'' assigns all values from 0 to ''p''-1 to the vertices.  A graph ''G'' is said to be edge-graceful if it admits an edge-graceful labeling.
 
Edge-graceful labelings were first introduced by S. Lo in 1985.<ref>{{cite journal | zbl=0597.05054 |last=Lo | first=Sheng-Ping | title=On edge-graceful labelings of graphs |
conference=Proc. Conf., Sundance/Utah 1985 | journal=Congressus Numerantium | volume=50 | year=1985 | pages=231–241 }}</ref>
 
A necessary condition for a graph to be edge-graceful is '''Lo's condition''':
:q(q+1) =p/(p-1)2 mod p
 
===Harmonious labelings===
A '''harmonious labeling''' on a graph ''G'' is an injection from the vertices of ''G'' to the [[Group (mathematics)|group]] of integers [[Modular arithmetic|modulo]] ''k'', where ''k'' is the number of edges of ''G'', that induces a bijection between the edges of ''G'' and the numbers modulo ''k'' by taking the edge label for an edge ''xy'' to be the sum of the labels of the two vertices ''x'', ''y'' (mod ''k''). A '''harmonious graph''' is one that has a harmonious labeling.  Odd cycles are harmonious, as is the [[Petersen graph]].  It is conjectured that trees are all harmonious if one vertex label is allowed to be reused.<ref>Guy (2004) pp.190–191</ref>
 
===Graph coloring===
[[Graph coloring]]s form a subclass of graph labelings.  A ''vertex coloring'' assigns different labels to adjacent vertices; an ''edge colouring'' different labels to adjacent edges.
 
===Lucky labeling===
A '''lucky labeling''' of a graph ''G'' is an assignment of positive integers to the vertices of ''G'' such that if ''S''(''v'') denotes the sum of the labels on the neighbours of ''v'', then ''S'' is a vertex coloring of ''G''. The '''lucky number''' of ''G'' is the least ''k'' such that ''G'' has a lucky labeling with the integers {1,...,''k''}.<ref>{{cite journal | zbl=1197.05125 | last1=Czerwiński | first1=Sebastian | last2=Grytczuk | first2=Jarosław | last3=Ẓelazny | first3=Wiktor | title=Lucky labelings of graphs | journal=Inf. Process. Lett. | volume=109 | number=18 | pages=1078–1081 | year=2009 }}</ref>
 
==References==
{{Reflist}}
* {{cite conference | zbl=0193.53204 | last=Rosa | first=A. | title=On certain valuations of the vertices of a graph | conference=Theory of Graphs, Int. Symp. Rome July 1966 | pages=349–355 | year=1967 | publisher=Gordon and Breach }}
* [[Joseph Gallian|Gallian, Joseph A.]] "A Dynamic Survey of Graph Labeling." The Electronic Journal of Combinatorics (2005). 20 Dec. 2006 [http://www.combinatorics.org/Surveys/ds6.pdf].
* {{cite book | last=Guy | first=Richard K. | authorlink=Richard K. Guy | title=Unsolved problems in number theory | publisher=[[Springer-Verlag]] |edition=3rd | year=2004 |isbn=0-387-20860-7 | at=C13 | zbl=1058.11001 }}
 
[[Category:Extensions and generalizations of graphs]]

Revision as of 18:39, 8 November 2013

In the mathematical discipline of graph theory, a graph labeling is the assignment of labels, traditionally represented by integers, to the edges or vertices, or both, of a graph.[1]

Formally, given a graph G, a vertex labeling is a function of G to a set of labels. A graph with such a function defined is called a vertex-labeled graph. Likewise, an edge labeling is a function mapping edges of G to a set of "labels". In this case, G is called an edge-labeled graph. When the edge labels are members of an ordered set (e.g., the real numbers), it may be called a weighted graph.

When used without qualification, the term labeled graph generally refers to a vertex-labeled graph with all labels distinct. Such a graph may equivalently be labeled by the consecutive integers {1, ..., n}, where n is the number of vertices in the graph.[1] For many applications, the edges or vertices are given labels that are meaningful in the associated domain. For example, the edges may be assigned weights representing the "cost" of traversing between the incident vertices.[2]

In the above definition a graph is understood to be a finite undirected simple graph. However, the notion of labeling may be applied to all extensions and generalizations of graphs. For example, in automata theory and formal language theory it is convenient to consider labeled multigraphs, i.e., a pair of vertices may be connected by several labeled edges.[3]

History

Most graph labelings trace their origins to labelings presented by Alex Rosa in his 1967 paper.[4] Rosa identified three types of labelings, which he called α-, β-, and ρ-labelings.[5] β-Labelings were later renamed graceful by S.W. Golomb and the name has been popular since.

Special cases

Graceful labeling

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A graceful labeling. Vertex labels are in black, edge labels in red

A graph is known as graceful when its vertices are labeled from 0 to , the size of the graph, and this labeling induces an edge labeling from 1 to . For any edge e, e's label is the positive difference between the two vertices incident with e. In other words, if e is incident with vertices labeled k and j, e will be labeled . Thus, a graph is graceful if and only if there exists an injection that induces a bijection from E to the positive integers up to .

In his original paper, Rosa proved that all eulerian graphs with order equivalent to 1 or 2 (mod 4) are not graceful. Whether or not certain families of graphs are graceful is an area of graph theory under extensive study. Arguably, the largest unproven conjecture in Graph Labeling is the Ringel-Kotzig conjecture, which hypothesizes that all trees are graceful. This has been proven for all paths, caterpillars, and many other infinite families of trees. Kotzig himself has called the effort to prove the conjecture a "disease."[6]

Edge-graceful labeling

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. An edge-graceful labeling on a simple graph (no loops or multiple edges) on p vertices and q edges is a labelling of the edges by distinct integers in {1,...,q} such that the labeling on the vertices induced by labeling a vertex with the sum of the incident edges taken modulo p assigns all values from 0 to p-1 to the vertices. A graph G is said to be edge-graceful if it admits an edge-graceful labeling.

Edge-graceful labelings were first introduced by S. Lo in 1985.[7]

A necessary condition for a graph to be edge-graceful is Lo's condition:

q(q+1) =p/(p-1)2 mod p

Harmonious labelings

A harmonious labeling on a graph G is an injection from the vertices of G to the group of integers modulo k, where k is the number of edges of G, that induces a bijection between the edges of G and the numbers modulo k by taking the edge label for an edge xy to be the sum of the labels of the two vertices x, y (mod k). A harmonious graph is one that has a harmonious labeling. Odd cycles are harmonious, as is the Petersen graph. It is conjectured that trees are all harmonious if one vertex label is allowed to be reused.[8]

Graph coloring

Graph colorings form a subclass of graph labelings. A vertex coloring assigns different labels to adjacent vertices; an edge colouring different labels to adjacent edges.

Lucky labeling

A lucky labeling of a graph G is an assignment of positive integers to the vertices of G such that if S(v) denotes the sum of the labels on the neighbours of v, then S is a vertex coloring of G. The lucky number of G is the least k such that G has a lucky labeling with the integers {1,...,k}.[9]

References

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  • Gallian, Joseph A. "A Dynamic Survey of Graph Labeling." The Electronic Journal of Combinatorics (2005). 20 Dec. 2006 [1].
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  1. ↑ 1.0 1.1 22 year-old Systems Analyst Rave from Merrickville-Wolford, has lots of hobbies and interests including quick cars, property developers in singapore and baking. Always loves visiting spots like Historic Monuments Zone of Querétaro.

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  2. ↑ "Different Aspects of Coding Theory", by Robert Calderbank (1995) ISBN 0-8218-0379-4, p. 53"
  3. ↑ "Developments in Language Theory", Proc. 9th. Internat.Conf., 2005, ISBN 3-540-26546-5, p. 313
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    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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  5. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

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    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

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    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  7. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  8. ↑ Guy (2004) pp.190–191
  9. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
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