# How do you prove 3 coloring is NP-complete?

## How do you prove 3 coloring is NP-complete?

To conclude, weve shown that 3-COLOURING is in NP and that it is NP-hard by giving a reduction from 3-SAT. Therefore 3-COLOURING is NP-complete.

## What is the three coloring problem?

This issue is a part of graph theory. It is well known that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color.

Is 4 coloring NP-complete?

This reduction takes linear time to add a single node and ¥ edges. Since 4-COLOR is in NP and NP-hard, we know it is NP-complete.

Is K coloring NP-complete?

Theorem: Independent set is NP-complete. A k-coloring of an undirected graph G is an assignment of colors to nodes such that each node is assigned a different color from all its neighbors, and at most k colors are used. Theorem: 3-COLORING is NP-Complete.

### Is 3 Colour NP-complete?

The 3-coloring problem remains NP-complete even on 4-regular planar graphs. However, for every k > 3, a k-coloring of a planar graph exists by the four color theorem, and it is possible to find such a coloring in polynomial time.

### What is a 3-coloring graph?

An instance of the 3-coloring problem is an undirected graph G (V, E), and the task is to check whether there is a possible assignment of colors for each of the vertices V using only 3 different colors with each neighbor colored differently.

Is map coloring NP-hard?

In particular, it is NP-hard to compute the chromatic number. The 3-coloring problem remains NP-complete even on 4-regular planar graphs. However, for every k > 3, a k-coloring of a planar graph exists by the four color theorem, and it is possible to find such a coloring in polynomial time.

Does co NP have 3 colors?

k-coloring asks if the nodes of a graph can be colored with ≤ k colors such that no two adjacent nodes have the same color. But 3-coloring is NP-complete (see next page).

## Is 2 coloring in the class NP-complete?

Since graph 2-coloring is in P and it is not the trivial language (∅ or Σ∗), it is NP-complete if and only if P=NP.

## Is 2-coloring in the class NP-complete?

How do you prove clique is NP?

Proof. 1. To show CLIQUE is in NP, our verifier takes a graph G(V,E), k, and a set S and checks if |S| ≥ k then checks whether (u, v) ∈ E for every u, v ∈ S. Thus the verification is done in O(n2) time.

Can a 3 SAT problem be transformed into a 3 coloring problem?

Show that any 3-SAT problem can be transformed into a 3-coloring problem in polynomial time. This equivalence is usually the harder part. In this case, one needs to show that, given some instance of a 3-SAT problem, we can get a graph s.t. that graph is 3-colorable if and only if the instance is satisfiable.

### Is the k coloring problem a SAT problem?

The k -coloring problem is to color any graph. You can certainly find graphs for which k -coloring is trivial as well as formulas for which SAT is trivial or etc. This does not impact the complexity of the problems in general though. Thanks for contributing an answer to Computer Science Stack Exchange!

### How to understand the reduction from 3-coloring problem to N colourability?

But the n -colourability problem for any constant n is the problem of determining whether an arbitrary input graph, with any number of vertices, has a proper n -colouring. The chain of reductions from 3 -colourability to n -colourability adds n − 3 vertices to the graph.

What is the statement for the 3 color problem?

As I understand, the statement for the 3-COLOR problem is:”Given any graph X, determine if it’s 3-colorable”. The process to arrive at the contradiction works if we can find a 3-CNF that can be reduced to X, or equivalently, if the graph G as defined above and corresponding to that 3-CNF is isomorphic to X.