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.

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.

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

Theorem: 3-COLORING is NP-Complete. Proof: (1) In NP: witness is a 3-coloring. We would like T, F, and R to be forced to different colors, so we will add edges between them to form a triangle. For the remaining nodes, and node that is colored the same color as T/F/R will be called colored TRUE/FALSE/RED, respectively.

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. Recall that by our hypothesis d(x) ≥ 8.

Can any graph be colored with 3 colors?

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 chromatic number explain with example?

The chromatic number of a graph is the smallest number of colors needed to color the vertices of so that no two adjacent vertices share the same color (Skiena 1990, p. 210), i.e., the smallest value of. possible to obtain a k-coloring. Minimal colorings and chromatic numbers for a sample of graphs are illustrated above …

Is Travelling salesman problem 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). I thought for A to be reduced to B, B has to be as hard if not harder than A.

Why is knapsack problem NP-hard?

The knapsack problem is NP-complete because the known NP-complete problem subset-sum is polynomially reducible to the knapsack problem, hence every problem in is reducible to the knapsack problem.

Is 4 coloring NP-complete?

Since 4-COLOR is in NP and NP-hard, we know it is NP-complete.

What is Hamiltonian cycle with example?

A dodecahedron ( a regular solid figure with twelve equal pentagonal faces) has a Hamiltonian cycle. A Hamiltonian cycle is a closed loop on a graph where every node (vertex) is visited exactly once.

Is g3 colorable?

Since G is 3-colorable, the neighbors of any vertex v form a bipartite graph (since none of these vertices can have the same color as vertex v). Thus, we can color the set δ(v) using two colors.

What is Welsh Powell algorithm?

Welsh Powell Algorithm consists of following Steps : Find the degree of each vertex. List the vertices in order of descending degrees. Colour the first vertex with color 1. Move down the list and color all the vertices not connected to the coloured vertex, with the same color.