3SAT23Colorability

3SAT23Colorability - Theorem1.3ColorabilityisNPComplete.

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Theorem 1.  3–Colorability is NP–Complete. Proof:  Given a YES instance of 3–Colorability and a 3-coloring of the vertices is  easy, in polynomial time, to verify the instance is YES. Thus 3-Coloribility is in the  set NP. In order to transform 3SAT to 3-colorability, we have to construct a  graph  of  node s and  edge s which is  3-colorable  if and only if the given instance of  3SAT   can be satisfied. This means we need a  graph  that can be 3-colored only when a  corresponding expression evaluates to true. For this example, the three colors  are  TRUE FALSE  and  OTHER . To construct the graph, start with a node named  root  that is colored  OTHER . Also, for every  variable   x  in the expression, create a  pair of nodes  x  and  ¬x . This pair is connected to each other and to the  root , to  create a  triangle  for each variable:
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3SAT23Colorability - Theorem1.3ColorabilityisNPComplete.

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