Tutte theorem

Tutte theorem

Let

G

⁢

(

V

,

E

)

be any

finite graph

.

G

has a

complete matching

if and only if for all

X

⊆

V

⁢

(

G

)

the inequality

c

p

⁢

(

G

-

X

)

≤

|

X

|

holds, where

c

p

⁢

(

H

)

is the number of the

components

of the

H

graph with

odd number

of vertices. This is called the Tutte condition.

Proof.

In a complete matching there is at least one edge between

X

and the odd components of

G

-

X

, and different edges have distinct

endvertices

. So

c

p

⁢

(

G

-

X

)

≤

|

X

|

. This proves the

necessity

.

The sufficiency is proved in a few steps indirectly.

1.

It is very easy to see that if

|

V

⁢

(

G

)

|

is even, then

c

p

⁢

(

G

-

X

)

≠

|

X

|

-

1

and

c

p

⁢

(

G

-

X

)

≠

|

X

|

+

1

. This will be used later. And if the Tutte condition holds, applying it to

X

=

∅

gives

|

V

⁢

(

G

)

|

is even.

2.

Now assume there exists a graph satisfying the Tutte condition but without complete matching ,and let

G

be such a

counterexample

with the lowest number of vertices. There exist a set

Y

⊆

V

⁢

(

G

)

such that

c

p

⁢

(

G

-

Y

)

=

|

Y

|

, examples are the

empty set

and every vertex. Let

Y

0

be such set with the highest number of vertices.

3.

If

G

s

is an even component in

G

-

Y

0

, then adding its vertex

t

to

Y

0

gives

c

p

⁢

(

G

-

Y

0

-

{

t

}

)

≥

|

Y

0

|

+

1

, because it creates at least one odd component. But this with the Tutte condition gives

c

p

⁢

(

G

-

Y

0

-

{

t

}

)

=

|

Y

0

∪

{

t

}

|

, which contradicts the maximality of

Y

0

.
So there are no even components in

G

-

Y

0

.

4.

Let

G

p

be an odd component of

G

-

Y

0

, and let

t

any of its vertices. Assume there is no complete matching in

G

p

-

{

t

}

. Since

G

is minimal counterexample, there exits a set

X

⊆

G

p

-

{

t

}

such that

c

p

⁢

(

G

p

-

{

t

}

-

X

)

>

|

X

|

, but because of

(

1

)

it is at least

|

X

|

+

2

. Then

c

p

⁢

(

G

-

X

-

Y

0

-

{

t

}

)

≥

|

Y

0

|

-

1

+

|

X

|

+

2

=

|

X

∪

Y

0

∪

{

t

}

|

, which contradicts the maximality of

Y

0

. So by removing any vertex from an odd component the remaining part has a complete matching.

5.

Represent each odd components by one vertex, and remove the edges in

Y

0

. From the

definition

of

Y

0

this is a

bipartite graph

. If we choose

p

vertices from the representatives of the odd components, they are together adjacent to at least

p

vertices in

Y

0

, otherwise we could remove less than

p

vertices from

Y

0

thus from

V

⁢

(

G

)

, and still get

p

odd component, which violates the Tutte condition. So there is complete matching in this

reduced

graph because of Hall’s

marriage theorem

.

6.

The complete matching in the reduced graph covers

Y

0

and exactly one vertex in each odd component. But there is also a

matching

in the rest of each odd component covering the rest as proven in

(

4

)

, and from

(

3

)

there are no even components. Combining these gives a complete matching of

G

, which is a

contradiction

.

∎

Title

Tutte theorem

Canonical

name

TutteTheorem

Date of creation

2013-03-22 14:06:04

Last modified on

2013-03-22 14:06:04

Owner

scineram (4030)

Last modified by

scineram (4030)

Numerical id

11

Author

scineram (4030)

Entry type

Theorem

Classification

msc 05C70

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