Reduce the
NOT-ALL-EQUAL-3-SAT problem to the
3-COLORABILITY problem.
These problems are defined as follows:
- NOT-ALL-EQUAL-3-SAT: given a boolean formula $F$ in conjunctive normal
form where each clause has $3$ literals, is there a truth assignment to the
variables of $F$ such that each clause of $F$ has at least one true literal and
one false literal?
More formally, this problem can be defined as the following set:
$\{ F=C_1\wedge\ldots\wedge C_n \mid F\text{ satisfiable with at least one true literal and one false literal per clause}\;\wedge\;\forall i\in\{1,\ldots,n\}: |C_i|=3 \}$
- 3-COLORABILITY: given an undirected graph $G$, is $G$ $3$-colorable?
In other words, is there a mapping $M$ from the nodes of $G$ to $\{1,2,3\}$
satisfying $M(u)\neq M(v)$ for each edge $\{u,v\}$ of $G$?
More formally, this problem can be defined as the following set:
$\{ G=\langle V,E\rangle \mid \exists M:V\to\{1,2,3\}\text{ total}: \forall \{u,v\}\in E: M(u)\neq M(v) \}$
The input and output of the reduction conform to the following data types: