## What is meant by coloring graph?

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Graph coloring is the procedure of assignment of colors to each vertex of a graph G such that no adjacent vertices get same color. The objective is to minimize the number of colors while coloring a graph. The smallest number of colors required to color a graph G is called its chromatic number of that graph.

**What is graph coloring in algorithm?**

Graph coloring algorithms. Given an undirected graph, a graph coloring is an assignment of labels traditionally called “colors” to each vertex. A graph coloring must have a special property: given two adjacent vertices, i.e., such that there exists an edge between them, they must not share the same color.

**What is graph coloring problem explain with example?**

Graph coloring problem involves assigning colors to certain elements of a graph subject to certain restrictions and constraints. In other words, the process of assigning colors to the vertices such that no two adjacent vertexes have the same color is caller Graph Colouring. This is also known as vertex coloring.

### What is the time complexity of graph coloring problem using a backtracking?

Time Complexity: O(mV). Since backtracking is also a kind of brute force approach, there would be total O(mV) possible color combinations.

**What are the properties of graph coloring?**

A coloring is proper if adjacent vertices have different colors. A graph is k-colorable if there is a proper k-coloring. The chromatic number χ(G) of a graph G is the minimum k such that G is k-colorable. Let H and G be graphs.

**What is the definition of graph according to graph theory?**

Definition: Graph is a mathematical representation of a network and it describes the relationship between lines and points. A graph consists of some points and lines between them. The length of the lines and position of the points do not matter. Each object in a graph is called a node.

#### Who introduced graph coloring?

Arthur Cayley raised the problem at a meeting of the London Mathematical Society in 1879. The same year, Alfred Kempe published a paper that claimed to establish the result, and for a decade the four color problem was considered solved.

**How graph coloring is solved through backtracking explain?**

Using Backtracking Algorithm In this approach, we color a single vertex and then move to its adjacent (connected) vertex to color it with different color. After coloring, we again move to another adjacent vertex that is uncolored and repeat the process until all vertices of the given graph are colored.

**What is time complexity and space complexity?**

Time complexity of an algorithm quantifies the amount of time taken by an algorithm to run as a function of the length of the input. Similarly, Space complexity of an algorithm quantifies the amount of space or memory taken by an algorithm to run as a function of the length of the input.

## What is a graph coloring problem?

Graph coloring problem is to assign colors to certain elements of a graph subject to certain constraints. Vertex coloring is the most common graph coloring problem.

**Is there any efficient algorithm for coloring a graph with minimum colors?**

Unfortunately, there is no efficient algorithm available for coloring a graph with minimum number of colors as the problem is a known NP Complete problem. There are approximate algorithms to solve the problem though. Following is the basic Greedy Algorithm to assign colors.

**What is the history of graph coloring?**

Graph coloring has been studied as an algorithmic problem since the early 1970s: the chromatic number problem (see below) is one of Karp’s 21 NP-complete problems from 1972, and at approximately the same time various exponential-time algorithms were developed based on backtracking and on the deletion-contraction recurrence of Zykov (1949).

### What is the color set of a graph?

In general, one can use any finite set as the “color set”. The nature of the coloring problem depends on the number of colors but not on what they are. Graph coloring enjoys many practical applications as well as theoretical challenges.