34-3 Graph coloring

Mapmakers try to use as few colors as possible when coloring countries on a map, as long as no two countries that share a border have the same color. We can model this problem with an undirected graph $G = (V, E)$ in which each vertex represents a country and vertices whose respective countries share a border are adjacent. Then, a $k$-coloring is a function $c: V \to \{1, 2, \dots, k \}$ such that $c(u) \ne c(v)$ for every edge $(u, v) \in E$. In other words, the numbers $1, 2, \dots, k$ represent the $k$ colors, and adjacent vertices must have different colors. The graph-coloring problem is to determine the minimum number of colors needed to color a given graph.

a. Give an efficient algorithm to determine a $2$-coloring of a graph, if one exists.

b. Cast the graph-coloring problem as a decision problem. Show that your decision problem is solvable in polynomial time if and only if the graph-coloring problem is solvable in polynomial time.

c. Let the language $\text{3-COLOR}$ be the set of graphs that can be $3$-colored. Show that if $\text{3-COLOR}$ is $\text{NP-complete}$, then your decision problem from part (b) is $\text{NP-complete}$.

To prove that $\text{3-COLOR}$ is $\text{NP-complete}$, we use a reduction from $\text{3-CNF-SAT}$. Given a formula $\phi$ of $m$ clauses on $n$ variables $x_1, x_2, \dots, x_n$, we construct a graph $G = (V, E)$ as follows. The set $V$ consists of a vertex for each variable, a vertex for the negation of each variable, $5$ vertices for each clause, and $3$ special vertices: $\text{TRUE}$, $\text{FALSE}$, and $\text{RED}$. The edges of the graph are of two types: "literal" edges that are independent of the clauses and "clause" edges that depend on the clauses. The literal edges form a triangle on the special vertices and also form a triangle on $x_i, \neg x_i$, and $\text{RED}$ for $i = 1, 2, \dots, n$.

d. Argue that in any $3$-coloring $c$ of a graph containing the literal edges, exactly one of a variable and its negation is colored $c(\text{TRUE})$ and the other is colored $c(\text{FALSE})$. Argue that for any truth assignment for $\phi$, there exists a $3$-coloring of the graph containing just the literal edges.

The widget shown in Figure 34.20 helps to enforce the condition corresponding to a clause $(x \vee y \vee z)$. Each clause requires a unique copy of the $5$ vertices that are heavily shaded in the figure; they connect as shown to the literals of the clause and the special vertex $\text{TRUE}$.

e. Argue that if each of $x$, $y$, and $z$ is colored $c(\text{TRUE})$ or $c(\text{FALSE})$, then the widget is $3$-colorable if and only if at least one of $x$, $y$, or $z$ is colored $c(\text{TRUE})$.

f. Complete the proof that $\text{3-COLOR}$ is $\text{NP-complete}$.