Question

A k-coloring of an undirected graph G = (V,E) is an assignment of a color to each vertex where every color is from the set {1, 2, ..., k}, such that for every edge (u, v) E E, the endpoints u and v are assigned different colors. Here, k is an integer greater than 0.

Consider the k-COLORING problem defined as: Input: An undirected graph G = (V,E) and an integer k>0
Output: A k-coloring of G if one exists, else output NO.
Using the fact that 3-COLORING is NP-complete, prove that 4-COLORING is NP-complete.

Answer 1

To prove that 4-COLORING is NP-complete, we can reduce the 3-COLORING problem to the 4-COLORING problem.

Step-by-step explanation:

To prove that 4-COLORING is NP-complete, we can reduce the 3-COLORING problem to the 4-COLORING problem.

We start with an instance of the 3-COLORING problem, which is an undirected graph G = (V, E) where each vertex can be assigned one of three colors such that no two adjacent vertices have the same color.

We then construct a new graph G' by adding a new vertex connected to all the vertices in G. This new vertex forces all the other vertices to have a different color, effectively solving the 4-COLORING problem.