Is 3-coloring NP-hard?

But this output node is adjacent to the False vertex coloured F; thus contradicting the 3-colourability of G! 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.

Is coloring NP-complete?

The Graph Coloring decision problem is np-complete, i.e, asking for existence of a coloring with less than ‘q’ colors, as given a coloring , it can be easily checked in polynomial time, whether or not it uses less than ‘q’ colors.

What is the 3-coloring problem?

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.

What does NP-complete mean?

NP-complete problem, any of a class of computational problems for which no efficient solution algorithm has been found. If a problem is NP and all other NP problems are polynomial-time reducible to it, the problem is NP-complete.

Why is 3 coloring a NP complete problem?

Because in this case, the output of the OR-gadget graph for Cj has to be colored False. This is a contradiction because the output is connected to Base and False. Hence, there exists a satisfying assignment to the 3-SAT clause. Conclusion: Therefore, 3-coloring is an NP-Complete problem.

Is the graph k coloring problem NP complete?

Thus, it can be concluded that the Graph K-coloring Problem is NP-Complete using the following two propositions:

What is the problem statement for 3 coloring?

Problem Statement: Given a graph G (V, E) and an integer K = 3, the task is to determine if the graph can be colored using at most 3 colors such that no two adjacent vertices are given the same color. An instance of the problem is an input specified to the problem.

How to prove that the problem is NP-complete?

●This problem is known to be NP-complete by a reduction from 3SAT. 3COLOR ∈ NP ●We can prove that 3COLOR ∈ NPby designing a polynomial-time nondeterministic TM for 3COLOR. ●M = “On input ⟨G⟩: ●Nondeterministicallyguess an assignment of colors to the nodes. ●Deterministicallycheck whether it is a 3- coloring. ●If so, accept; otherwise reject.”