Is 3 Colorability 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 a 3-coloring graph?
Definition 1 A graph G is 3-colorable if the vertices of a given graph can be colored with only three colors, such that no two vertices of the same color are connected by an edge. Ie: Coloring all red vertices blue and coloring all blue vertices red gives a valid 3-coloring.
What is the three coloring problem?
The Three Color Problem is: Under what conditions can the regions of a planar map be colored in three colors so that no two regions with a common boundary have the same color? This paper describes the origin of the Three Color Problem and virtually all the major results and conjectures extant in the literature.
How do you know if a graph is three colorable?
Let x be a vertex in V (G) − (N[v] ∪ N2(v)). In any proper 3-coloring of G, if it exists, the vertex x either gets the same color as v or x receives a different color than v. Therefore it is enough to determine if any of the graphs G/xv and G ∪ xv are 3-colorable.
Is graph coloring NP-complete?
Vertex coloring of a graph is a well-known NP-complete problem, but for certain classes of graphs it can be solved in polynomial time [lo]. For example, the com- plements of transitively orientable (coTR0) graphs can be colored in 0(n4) time, where n is the number of vertices [5].
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 every graph 3-colorable?
Every planar graph without adjacent 3-cycles and without 5-cycles is 3-colorable.
Which of the following graph is not 3-colorable?
Almost all graphs with 2.522 n edges are not 3-colorable.
Is TSP NP-complete?
Traveling Salesman Optimization(TSP-OPT) is a NP-hard problem and Traveling Salesman Search(TSP) is NP-complete. However, TSP-OPT can be reduced to TSP since if TSP can be solved in polynomial time, then so can TSP-OPT(1).
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:
When is a NP problem called NP hard?
If the 2nd condition is only satisfied then the problem is called NP-Hard. But it is not possible to reduce every NP problem into another NP problem to show its NP-Completeness all the time.
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.