Question: What Is Chromatic Number In Graph Theory?

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.

Coloring here means attaching a “color” or a number to each vertice such that no two vertices with a connecting edge have the save value..

What are the application of graph Colouring?

The graph coloring problem has huge number of applications. 1) Making Schedule or Time Table: Suppose we want to make am exam schedule for a university. We have list different subjects and students enrolled in every subject. Many subjects would have common students (of same batch, some backlog students, etc).

What is graph coloring give an example?

Graph coloring is nothing but a simple way of labelling graph components such as vertices, edges, and regions under some constraints. In a graph, no two adjacent vertices, adjacent edges, or adjacent regions are colored with minimum number of colors. … A coloring is given to a vertex or a particular region.

Is a graph 2 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).

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(Sk),of a surface Sk is the largest χ(G) such that G can be imbedded in Sk. We prove that six colors will suffice for every planar graph.

How do you prove a graph is three colorable?

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. In other words given a graph we denote each vertices as vi and vj where i,j < n.

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.

What is the chromatic number of a cycle graph?

The number of vertices in Cn equals the number of edges, and every vertex has degree 2; that is, every vertex has exactly two edges incident with it….Cycle graphChromatic number3 if n is odd 2 otherwiseChromatic index3 if n is odd 2 otherwiseSpectrum{2 cos(2kπ/n); k = 1, …, n}8 more rows

What is the main idea of graph Colouring problem explain with example?

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 a K4 graph?

K4 is a maximal planar graph which can be seen easily. In fact, a planar graph G is a maximal planar graph if and only if each face is of length three in any planar embedding of G. Corollary 1.8. 2: The number of edges in a maximal planar graph is 3n-6.

Is a self loop a cycle?

A cycle in a graph is, according to Wikipedia, An edge set that has even degree at every vertex; also called an even edge set or, when taken together with its vertices, an even subgraph. … Therefore the self-loop is a cycle in your graph.

What is a cycle graph theory?

In graph theory, a cycle in a graph is a non-empty trail in which the only repeated vertices are the first and last vertices. A directed cycle in a directed graph is a non-empty directed trail in which the only repeated vertices are the first and last vertices. A graph without cycles is called an acyclic 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.

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.

What is the chromatic number of a tree?

Trees- A Tree is a special type of connected graph in which there are no circuits. Every tree is a bipartite graph. So, chromatic number of a tree with any number of vertices = 2.

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