# What is chromatic polynomial and chromatic number?

## What is chromatic polynomial and chromatic number?

The chromatic polynomial is called the “chromial” for short by Bari (1974). The chromatic number of a graph gives the smallest number of colors with which a graph can be colored, which is therefore the smallest positive integer such that. (Skiena 1990, p. 211).

What is the chromatic polynomial of a cycle?

Proposition The chromatic polynomial of the cycle of size n is (k-1) +(-1) (k-1). Proof We proceed by induction. Base case: n=2. This is the complete graph (plus a multiple edge), so the chromatic polynomial is k(k-1).

What is chromatic graph?

The chromatic number of a graph is the smallest number of colors needed to color the vertices of so that no two adjacent vertices share the same color (Skiena 1990, p. 210), i.e., the smallest value of. possible to obtain a k-coloring.

### Why is the chromatic polynomial a polynomial?

Because the Chromatic Function of a null graph is a polynomial (PNn (k) = kn), we see that the Chromatic Function of G is equal to the sum of a large number of polynomials and must itself be a polynomial.

What is the chromatic number of the following graph?

18) The chromatic number of the following graph is _________ . Explanation: The chromatic number of a graph is the minimum number of colors needed to produce a proper coloring of a graph and no two vertex which share same edge have the same color. Hence, the chromatic number of a given graph is 3.

What is the chromatic formula?

If G is a simple graph, we write PG(k) as the number of ways we can achieve a proper coloring on the vertices of G given k colors and PG is called the Chromatic Function of G. If k<χ(G), then PG(k) = 0.

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

In a complete graph, each vertex is adjacent to is remaining (n–1) vertices. Hence, each vertex requires a new color. Hence the chromatic number Kn = n.

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. Def. 8-2. 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 find the chromatic graph?

Color the currently picked vertex with the lowest numbered color if it has not been used to color any of its adjacent vertices. If it has been used, then choose the next least numbered color. If all the previously used colors have been used, then assign a new color to the currently picked vertex.

### What is the chromatic number of a complete graph?

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.

When does a graph have a chromatic polynomial?

A cycle or a loop is when the graph is a path which close on itself. That mean that: Where E is the number of Edges and V the number of Vertices. The Chromatic Polynomial formula is: Where n is the number of Vertices.

## Who is the creator of the chromatic polynomial?

The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a function of the number of colors and was originally defined by George David Birkhoff to attack the four color problem.

How to calculate the chromatic polynomial in Python?

The Chromatic Polynomial formula is: Where n is the number of Vertices. Python Code: def chromatic_polynomial (lambda, vertices): return ( lambda – 1 ) ** vertices + ( ( -1 ) ** vertices) * ( lambda – 1 )

Which is a common problem in graph theory?

1. Introduction A common problem in the study of Graph Theory is coloring the vertices of a graph so that any two connected by a common edge are di\erent colors. The vertices of the graph in Figure 1 have been colored in the desired manner.

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