Graph 3-Colorability
Instance: a graph 𝐺 = (𝑉, 𝐸).
Question: Is there a color assignment
𝑐: 𝑉 → {𝟏, 𝟐, 𝟑}
such that 𝑐(𝑣𝑖 ) ≠ 𝑐(𝑣𝑗 ) if 𝑣𝑖 and 𝑣𝑗 are adjacent?
Example 4: 3-SAT → Graph 3-Colorability
Given an arbitrary instance of 3-Sat, (i.e. a set of variables 𝑈 = {𝑢1 , 𝑢2 , ⋯ , 𝑢𝑛 } and a set of size
three clauses, = {𝑐1 , 𝑐2 , ⋯ , 𝑐𝑚 } ) we must construct an equivalent instance of 3-Colorability.
The construction will be an example of “component design”. Each component will be a
subgraph that effectively performs the particular role.
Some of the vertices of the graph will correspond to variables. The way those vertices are
colored will determine a true/false assignment for the corresponding variables.
Example 4: 3-SAT → Graph 3-Colorability
Given an arbitrary instance of 3-Sat, (i.e. a set of variables 𝑈 = {𝑢1 , 𝑢2 , ⋯ , 𝑢𝑛 } and a set of size
three clauses, = {𝑐1 , 𝑐2 , ⋯ , 𝑐𝑚 } ) we must construct an equivalent instance of 3-Colorability.
1. Color-to-True/False Correspondence
The “center” of our graph will be the triangle below. These vertices do not
correspond to variables but one have the color corresponding to “true”, one will have the
color corresponding to “false” and the third will be the “neutral” or “neither” or “not like
the others” color. We can assume these vertices are colored as below.
F
T
N
Example 4: 3-SAT → Graph 3-Colorability
Given an arbitrary instance of 3-Sat, (i.e. a set of variables 𝑈 = {𝑢1 , 𝑢2 , ⋯ , 𝑢𝑛 } and a set of size
three clauses, = {𝑐1 , 𝑐2 , ⋯ , 𝑐𝑚 } ) we must construct an equivalent instance of 3-Colorability.
𝑢̅𝑛
F
⋰
𝑢̅2
T
N
𝑢𝑛
𝑢2
𝑢̅1
𝑢1
2. True/False Determination
Now we create 𝑛 pairs of new vertices. One vertex of the 𝑖 𝑡ℎ pair will correspond to
variable 𝑢𝑖 and the other vertex will correspond to 𝑢̅𝑖 . Connect each pair to the “N”
vertex above to form a subgraph that ensures that each vertex corresponding to a
variable gets colored either blue = “T” or “red = F” and that the vertex
corresponding to its negation is colored the opposite.
Side Note:
Suppose that the edges leading away have forced the green and red colors as shown.
Can the rest of the graph on left be 3-colored? How about the graph on the right?
Example 4: 3-SAT → Graph 3-Colorability
Given an arbitrary instance of 3-Sat, (i.e. a set of variables 𝑈 = {𝑢1 , 𝑢2 , ⋯ , 𝑢𝑛 } and a set of size
three clauses, = {𝑐1 , 𝑐2 , ⋯ , 𝑐𝑚 } ) we must construct an equivalent instance of 3-Colorability.
3. Satisfaction Testing
We will create one of the gadgets to the right
for each clause, 𝑐𝑘 . The vertices shown only appear within
one gadget. Arrows indicate connections to vertices
already introduced in above components.
We must find what the connections need to be so that the subgraph is 3-colorable
if and only if the clause is satisfied.