# How do you find the chromatic number of a graph?

## How do you find the chromatic number of a graph?

The minimum number of colors in a proper coloring of a graph G is called the (vertex) chromatic number of G and is denoted by χ(G). The chromatic number of many special graphs is easy to determine. For example, χ(Kn) = n, χ(Cn) = 3 if n is odd, and χ(B) = 2 for any bipartite graph B with at least one edge.

What is the chromatic number of Grotzsch graph?

In the mathematical field of graph theory, the Grötzsch graph is a triangle-free graph with 11 vertices, 20 edges, chromatic number 4, and crossing number 5.

What is the chromatic number of the following graphs?

Explanation: Chromatic number of given graph is 3.

### How many faces does Petersen graph have?

The Petersen graph can be embedded in the real projective plane with 6 faces (as the quotient of a dodecahedron by the antipodal map), or on the torus with 5 faces.

What is chromatic number of a graph explain with example?

The chromatic number of a graph is the minimum number of colors needed to produce a proper coloring of a graph. In our scheduling example, the chromatic number of the graph would be the minimum number of time slots needed to schedule the meetings so there are no time conflicts.

What is chromatic number of K5?

In this paper, we offer the following partial result: The chromatic number of a random lift of K5 \ e is a.a.s. three. We actually prove a stronger statement where K5 \ e can be replaced by a graph obtained from joining a cycle to a stable set.

#### What is the independence number of the Petersen graph?

4
The Petersen graph has independence number 4 and chromatic number 3.

What is the chromatic number of following number?

What will be the chromatic number of the following graph? Explanation: The given graph will only require 2 unique colors so that no two vertices connected by a common edge will have the same color. So its chromatic number will be 2.

What is the chromatic number of K3 3?

Chromatic polynomial for K3, 3 is given by λ(λ – 1)5. Thus chromatic number of this graph is 2. Since λ(λ – 1)5 > 0 first when λ = 2. Here, only two distinct colours are required to colour K3, 3.

## Is Petersen graph complete graph?

The Petersen graph is a particular undirected graph on 10 vertices that can be defined in the following equivalent ways: It is the complement of the line graph of complete graph:K5. It is the odd graph with parameter 3, i.e., the graph .

How many cycles does Petersen graph have?

Petersen Graph

property value
Hamiltonian graph no
Hamiltonian cycle count 0
Hamiltonian path count 240
hypohamiltonian graph yes

What is the chromatic index of a Petersen graph?

The Petersen graph has chromatic number 3, meaning that its vertices can be colored with three colors — but not with two — such that no edge connects vertices of the same color. It has a list coloring with 3 colors, by Brooks’ theorem for list colorings. The Petersen graph has chromatic index 4; coloring the edges requires four colors.

### What are the characteristics of the Petersen graph?

The Petersen graph has girth 5, diameter 2, edge chromatic number 4, chromatic number 3, and chromatic polynomial The Petersen graph is a cubic symmetric graph and is nonplanar.

Does the Petersen graph have a Hamiltonian cycle?

Since the Petersen graph has girth five, it cannot be formed in this way and has no Hamiltonian cycle. The Petersen graph has chromatic number 3, meaning that its vertices can be colored with three colors — but not with two — such that no edge connects vertices of the same color.

What is the chromatic number of an even cycle graph?

If number of vertices in cycle graph is even, then its chromatic number = 2. If number of vertices in cycle graph is odd, then its chromatic number = 3.