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Reduce

3SAT to 3COL, showing NP-completeness of the latter.

Perform a polynomial-time reduction using the graph below.

  1. Add three interconnected vertices, colored 1, 2, and 3.
  2. For each variable, add two uncolored vertices connected to 2.
  3. For each clause, add one Or gadget connected to:
    • the relevant variable vertices and
    • the vertex colored 1.

Given

F=c1c2=(111213)(212223),

connect the graph

svg/0e31ff94-c745-49ab-a42b-985b2a1ac01c

For example,

F=(¬x1x2¬x2)(¬x2x3¬x3)

reduces to the graph

svg/a2215ee0-36af-4e1f-8d8f-c8ff4fda71c5

© 2025 Rudolf Adamkovič under GNU General Public License version 3.
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