chromatic polynomial

chromatic polynomial

Let

G

be a graph (in the sense of

graph theory

) whose set

V

of vertices is finite and nonempty, and which has no loops or

multiple edges

.
For any

natural number

x

, let

χ

⁢

(

G

,

x

)

, or just

χ

⁢

(

x

)

,
denote the number of

x

-colorations of

G

, i.e. the number of

mappings

f

:

V

→

{

1

,

2

,

…

,

x

}

such that

f

⁢

(

a

)

≠

f

⁢

(

b

)

for any
pair

(

a

,

b

)

of

adjacent vertices

.
Let us prove that

χ

(which is called the

chromatic polynomial

of the graph

G

)
is a

polynomial function

in

x

with

coefficients

in

ℤ

.
Write

E

for the set of edges in

G

. If

|

E

|

=0, then trivially

χ

⁢

(

x

)

=

x

|

V

|

(where

|

⋅

|

denotes the number of elements
of a

finite set

).
If not, then we choose an edge

e

and construct two graphs
having fewer edges than

G

:

H

is obtained from

G

by contracting
the edge

e

, and

K

is obtained from

G

by omitting the edge

e

.
We have

χ

⁢

(

G

,

x

)

=

χ

⁢

(

K

,

x

)

-

χ

⁢

(

H

,

x

)

(1)

for all

x

∈

ℕ

, because the

polynomial

χ

⁢

(

K

,

x

)

is the number
of colorations of the vertices of

G

which might or might not be
valid for the edge

e

, while

χ

⁢

(

H

,

x

)

is the number which
are not valid.
By

induction

on

|

E

|

, (

1

) shows that

χ

⁢

(

G

,

x

)

is
a polynomial over

ℤ

.

By refining the

argument

a little, one can show

χ

⁢

(

x

)

=

x

|

V

|

-

|

E

|

⁢

x

|

V

|

-

1

+

…

±

s

⁢

x

k

,

for some nonzero integer

s

, where

k

is the number
of

connected components

of

G

, and the coefficients alternate in sign.

With the help of the Möbius-Rota

inversion

formula

(see Moebius Inversion),
or directly by induction, one can prove

χ

⁢

(

x

)

=

∑

F

⊂

E

(

-

1

)

|

F

|

⁢

x

|

V

|

-

r

⁢

(

F

)

where the sum is over all subsets

F

of

E

, and

r

⁢

(

F

)

denotes the
rank of

F

in

G

, i.e. the number of elements of
any maximal cycle-free subset of

F

.
(Alternatively, the sum may be taken only over subsets

F

such that

F

is equal to the span of

F

; all other summands cancel
out in pairs.)

The

chromatic number

of

G

is the smallest

x

>

0

such that

χ

⁢

(

G

,

x

)

>

0

or, equivalently, such that

χ

⁢

(

G

,

x

)

≠

0

.

The

Tutte polynomial

of a graph, or more generally of a

matroid

(

E

,

r

)

, is this function of two variables:

t

⁢

(

x

,

y

)

=

∑

F

⊂

E

(

x

-

1

)

r

⁢

(

E

)

-

r

⁢

(

F

)

⁢

(

y

-

1

)

|

F

|

-

r

⁢

(

F

)

.

Compared to the chromatic polynomial, the Tutte contains more information
about the matroid.
Still, two or more nonisomorphic matroids may have the same Tutte polynomial.

Title

chromatic polynomial

Canonical name

ChromaticPolynomial

Date of creation

2013-03-22 13:58:25

Last modified on

2013-03-22 13:58:25

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

8

Author

bbukh (348)

Entry type

Definition

Classification

msc 05C15

Classification

msc 05B35

Related topic

Matroid

Related topic

ChromaticNumber

Defines

Tutte polynomial

Generated on Thu Feb 8 20:42:15 2018 by

LaTeXML