# How to determine if a graph is 3-colorable?

Last Update: April 20, 2022

This is a question our experts keep getting from time to time. Now, we have got the complete detailed explanation and answer for everyone, who is interested!

**Asked by: Abdul Lubowitz**

Score: 4.5/5 (16 votes)

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.

## What makes a graph 3-colorable?

The graph 3-colorability problem is a decision problem in graph theory which asks **if it is possible to assign a color to each vertex of a given graph using at most three colors**, satisfying the condition that every two adjacent vertices have different colors.

## How do you know if a graph is two colorable?

A graph is 2-colorable if we can color each of its vertices with one of two colors, say red and blue, in such a way that no two red vertices are connected by an edge, and **no two blue vertices are connected by an edge** (a k-colorable graph is defined in a similar way).

## Is a graph n colorable?

Every graph with n vertices is n-colourable: **assign a different colour to every vertex**. Hence, there is a smallest k such that G is k-colourable.

## 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**.

## 6.3 Graph Coloring Problem - Backtracking

**32 related questions found**

### How do you prove a graph is K colorable?

A graph is k-colorable if **it is possible to assign each vertex to one of k colors such that the two endpoints of every edge are assigned different colors**.

### What is chromatic number of a graph explain with example?

The chromatic number, χ(G), of a graph G **is the smallest number of colors for V(G) so that adjacent vertices are colored differently**. The chromatic number, χ(S_{k}),of a surface S_{k} is the largest χ(G) such that G can be imbedded in S_{k}. We prove that six colors will suffice for every planar graph.

### What is the minimum number of colors required for any K partite graph?

It is trivially at least the chromatic number and can be that small; the dynamic chromatic number of a complete k-partite graph is k when k≥3 [LMP] (for a bipartite graph containing C_{4}, at least **four colors** are needed).

### What is meant by chromatic number of a graph?

(definition) Definition: **The minimum number of colors needed to color the vertices of a graph such that no two adjacent vertices have the same color**.

### Is every graph 2-colorable?

Upper bounds on the chromatic number

Finding cliques is known as the clique problem. The 2-colorable graphs are exactly the bipartite graphs, including trees and forests. By the four color theorem, **every planar graph can be 4-colored**. for a connected, simple graph G, unless G is a complete graph or an odd cycle.

### Why is coloring a graph necessary?

Actual colors have nothing at all to do with this, graph coloring is **used to solve problems where you have a limited amount of resources or other restrictions**. The colors are just an abstraction for whatever resource you're trying to optimize, and the graph is an abstraction of your problem.

### How do you prove a graph is bipartite?

**The graph is a bipartite graph if:**

- The vertex set of can be partitioned into two disjoint and independent sets and.
- All the edges from the edge set have one endpoint vertex from the set and another endpoint vertex from the set.

### Which of the following graph is not 3-colorable?

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

### How do you color graphs?

**Method to Color a Graph**

- Step 1 − Arrange the vertices of the graph in some order.
- Step 2 − Choose the first vertex and color it with the first color.
- Step 3 − Choose the next vertex and color it with the lowest numbered color that has not been colored on any vertices adjacent to it. ...
- Example.

### When exactly is a graph 2 Colourable?

More precisely, we shall color the vertices of a graph, observing two rules: **every vertex must be colored**, and two vertices linked by an edge cannot be given the same color. If n is a natural number, then a graph is said to be n-colorable if it can be colored using n different colors, but not with few colors than n.

### What is the fewest number of colors needed to color this graph?

Definition **16** (Chromatic Number). The chromatic number of a graph is the minimum number of colors in a proper coloring of that graph.

### What is a simple cycle?

A simple cycle is **a cycle with no repeated vertices (except for the beginning and ending vertex)**. Remark: If a graph contains a cycle from v to v, then it contains a simple cycle from v to v. ... Connected Graphs. A graph G is called connected if there is a path between any two distinct vertices of G.

### What is the vertex coloring of a graph?

Vertex coloring is **an assignment of colors to the vertices of a** graph 'G' such that no two adjacent vertices have the same color. Simply put, no two vertices of an edge should be of the same color.

### What is clique number in graph theory?

The clique cover number of a graph is **the smallest number of cliques of whose union covers the set of vertices**. **of the graph**. A maximum clique transversal of a graph is a subset of vertices with the property that each maximum clique of the graph contains at least one vertex in the subset.

### Can a spanning tree have cycles?

All the possible spanning trees of a graph have the same number of edges and vertices. **A spanning tree can never contain a cycle**. Spanning tree is always minimally connected i.e. if we remove one edge from the spanning tree, it will become disconnected.

### What is a digraph algorithm?

A directed graph (or digraph) is **a set of vertices and a collection of directed edges that each connects an ordered pair of vertices**. We say that a directed edge points from the first vertex in the pair and points to the second vertex in the pair. We use the names 0 through V-1 for the vertices in a V-vertex graph.

### What is MST in graph?

A **minimum spanning tree** (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. ... There are many use cases for minimum spanning trees.

### What is the condition for proper coloring of a graph?

What is the condition for proper coloring of a graph? Explanation: The condition for proper coloring of graph is that **two vertices which share a common edge should not have the same color**. If it uses k colors in the process then it is called k coloring of graph. 3.

### Is there a bipartite graph that is 1 colorable?

Theorem 2.7 (Bipartite Colorings) If G is a bipartite graph with a positive num- ber of edges, then G is 2-colorable. **If G is bipartite with no edges, it is 1-colorable**.