What is a 3-coloring graph?

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 3 color 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.

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.

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.

Why is NP 3 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 the 2 coloring problem in P or in NP?

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.

Can you color a map with 3 colors?

In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. Additionally, in 2005, the theorem was proved by Georges Gonthier with general-purpose theorem-proving software.

Are all graphs 3-colorable?

Every planar graph with neither 3-cycles at distance less than 4 nor 5-cycles is 3-colorable. Informally speaking, this result shows that the “intersection” of Havel’s and Steinberg’s problems has a positive solution, i.e., each graph satisfying both Havel’s and Steinberg’s assumptions is 3-colorable.

Is 2COLOR NP-hard?

In this algorithm each node will be used to try to color its neighbors only once, and 5 Page 6 each node has at most |V | neighbors, thus the algorithm runs in at most |V |2 time, which is polynomial in the length of the input. Thus, 2COLOR is in P. All of the problems within the complexity class NP.

Is every graph 3-colorable?

Every planar graph without adjacent 3-cycles and without 5-cycles is 3-colorable.

Which of the following graphs isnt 3-colorable?

Almost all graphs with 2.522 n edges are not 3-colorable.