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| [[File:Exact coloring.svg|thumb|right|Example of Exact Coloring with 7 colors and 14 vertices]]
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| 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.
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| 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
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| | last = Edwards | first = Keith
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| | doi = 10.1017/S0963548304006558
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| | issue = 3
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| | journal = Combinatorics, Probability and Computing
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| | mr = 2138114
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| | pages = 275–310
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| | title = Detachments of complete graphs
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| | volume = 14
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| | year = 2005}}.</ref><ref name="e10">{{citation
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| | last = Edwards | first = Keith
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| | doi = 10.1002/jgt.20468
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| | issue = 2
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| | journal = Journal of Graph Theory
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| | mr = 2724490
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| | pages = 94–114
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| | title = Achromatic number of fragmentable graphs
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| | volume = 65
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| | year = 2010}}.</ref>
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| ==Complete graphs, detachments, and Euler tours==
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| [[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.
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| 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"/>
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| 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"/>
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| ==Related types of coloring==
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| 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"/>
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| ==Computational complexity==
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| 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
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| | last1 = Edwards | first1 = Keith
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| | last2 = McDiarmid | first2 = Colin
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| | doi = 10.1016/0166-218X(94)00100-R
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| | issue = 2-3
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| | journal = Discrete Applied Mathematics
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| | mr = 1327772
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| | pages = 133–144
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| | title = The complexity of harmonious colouring for trees
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| | volume = 57
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| | 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
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| | last = Edwards | first = Keith
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| | doi = 10.1017/S0963548300001802
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| | issue = 1
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| | journal = Combinatorics, Probability and Computing
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| | mr = 1395690
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| | pages = 15–28
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| | title = The harmonious chromatic number of bounded degree trees
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| | volume = 5
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| | year = 1996}}.</ref>
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| ==References==
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| {{reflist}}
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| [[Category:Graph coloring]]
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