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[[File:Exact coloring.svg|thumb|right|Example of Exact Coloring with 7 colors and 14 vertices]] | |||
In [[graph theory]], an '''exact coloring''' is a [[Graph coloring|(proper) vertex coloring]] in which every pair of colors appears on exactly one pair of adjacent vertices. | |||
That is, it is a [[Partition of a set|partition]] of the vertices of the graph into disjoint [[independent set (graph theory)|independent sets]] such that, for each pair of distinct independent sets in the partition, there is exactly one edge with endpoints in each set.<ref name="e05">{{citation | |||
| last = Edwards | first = Keith | |||
| doi = 10.1017/S0963548304006558 | |||
| issue = 3 | |||
| journal = Combinatorics, Probability and Computing | |||
| mr = 2138114 | |||
| pages = 275–310 | |||
| title = Detachments of complete graphs | |||
| volume = 14 | |||
| year = 2005}}.</ref><ref name="e10">{{citation | |||
| last = Edwards | first = Keith | |||
| doi = 10.1002/jgt.20468 | |||
| issue = 2 | |||
| journal = Journal of Graph Theory | |||
| mr = 2724490 | |||
| pages = 94–114 | |||
| title = Achromatic number of fragmentable graphs | |||
| volume = 65 | |||
| year = 2010}}.</ref> | |||
==Complete graphs, detachments, and Euler tours== | |||
[[Image:Graph_exact_coloring.gif|thumb|Exact coloring of the [[complete graph]] ''K''<sub>6</sub>]] | |||
Every ''n''-vertex [[complete graph]] ''K''<sub>''n''</sub> has an exact coloring with ''n'' colors, obtained by giving each vertex a distinct color. | |||
Every graph with an ''n''-color exact coloring may be obtained as a ''detachment'' of a complete graph, a graph obtained from the complete graph by splitting each vertex into an independent set and reconnecting each edge incident to the vertex to exactly one of the members of the corresponding independent set.<ref name="e05"/><ref name="e10"/> | |||
When ''k'' is an [[odd number]], A path or cycle with <math>\tbinom{k}{2}</math> edges has an exact coloring, obtained by forming an exact coloring of the complete graph ''K''<sub>''k''</sub> and then finding an [[Euler tour]] of this complete graph. For instance, a path with three edges has a complete 3-coloring.<ref name="e10"/> | |||
==Related types of coloring== | |||
Exact colorings are closely related to [[harmonious coloring]]s (colorings in which each pair of colors appears at most once) and [[complete coloring]]s (colorings in which each pair of colors appears at least once). Clearly, an exact coloring is a coloring that is both harmonious and complete. A graph ''G'' with ''n'' vertices and ''m'' edges has a harmonious ''k''-coloring if and only if <math>m\le\tbinom{k}{2}</math> and the graph formed from ''G'' by adding <math>\tbinom{k}{2}-m</math> isolated edges has an exact coloring. A graph ''G'' with the same parameters has a complete ''k''-coloring if and only if <math>m\ge\tbinom{k}{2}</math> and there exists a subgraph ''H'' of ''G'' with an exact ''k''-coloring in which each edge of ''G'' − ''H'' has endpoints of different colorings. The need for the condition on the edges of ''G'' − ''H'' is shown by the example of a four-vertex cycle, which has a subgraph with an exact 3-coloring (the three-edge path) but does not have a complete 3-coloring itself.<ref name="e10"/> | |||
==Computational complexity== | |||
It is [[NP-complete]] to determine whether a given graph has an exact coloring, even in the case that the graph is a [[tree (graph theory)|tree]].<ref name="e05"/><ref>{{citation | |||
| last1 = Edwards | first1 = Keith | |||
| last2 = McDiarmid | first2 = Colin | |||
| doi = 10.1016/0166-218X(94)00100-R | |||
| issue = 2-3 | |||
| journal = Discrete Applied Mathematics | |||
| mr = 1327772 | |||
| pages = 133–144 | |||
| title = The complexity of harmonious colouring for trees | |||
| volume = 57 | |||
| year = 1995}}.</ref> However, the problem may be solved in [[polynomial time]] for trees of bounded [[degree (graph theory)|degree]].<ref name="e05"/><ref>{{citation | |||
| last = Edwards | first = Keith | |||
| doi = 10.1017/S0963548300001802 | |||
| issue = 1 | |||
| journal = Combinatorics, Probability and Computing | |||
| mr = 1395690 | |||
| pages = 15–28 | |||
| title = The harmonious chromatic number of bounded degree trees | |||
| volume = 5 | |||
| year = 1996}}.</ref> | |||
==References== | |||
{{reflist}} | |||
[[Category:Graph coloring]] | |||
Revision as of 07:02, 3 December 2013
In graph theory, an exact coloring is a (proper) vertex coloring in which every pair of colors appears on exactly one pair of adjacent vertices. That is, it is a partition of the vertices of the graph into disjoint independent sets such that, for each pair of distinct independent sets in the partition, there is exactly one edge with endpoints in each set.[1][2]
Complete graphs, detachments, and Euler tours
Every n-vertex complete graph Kn has an exact coloring with n colors, obtained by giving each vertex a distinct color. Every graph with an n-color exact coloring may be obtained as a detachment of a complete graph, a graph obtained from the complete graph by splitting each vertex into an independent set and reconnecting each edge incident to the vertex to exactly one of the members of the corresponding independent set.[1][2]
When k is an odd number, A path or cycle with edges has an exact coloring, obtained by forming an exact coloring of the complete graph Kk and then finding an Euler tour of this complete graph. For instance, a path with three edges has a complete 3-coloring.[2]
Related types of coloring
Exact colorings are closely related to harmonious colorings (colorings in which each pair of colors appears at most once) and complete colorings (colorings in which each pair of colors appears at least once). Clearly, an exact coloring is a coloring that is both harmonious and complete. A graph G with n vertices and m edges has a harmonious k-coloring if and only if and the graph formed from G by adding isolated edges has an exact coloring. A graph G with the same parameters has a complete k-coloring if and only if and there exists a subgraph H of G with an exact k-coloring in which each edge of G − H has endpoints of different colorings. The need for the condition on the edges of G − H is shown by the example of a four-vertex cycle, which has a subgraph with an exact 3-coloring (the three-edge path) but does not have a complete 3-coloring itself.[2]
Computational complexity
It is NP-complete to determine whether a given graph has an exact coloring, even in the case that the graph is a tree.[1][3] However, the problem may be solved in polynomial time for trees of bounded degree.[1][4]
References
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To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. - ↑ 2.0 2.1 2.2 2.3 Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. - ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. - ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.