aarjaneiro/planetmath huggingface dataset:
k-connected graph

k

-connected graph

Connectivity of graphs, when it isn’t specified which flavor is intended, usually refers to

vertex connectivity

, unless it is clear from the

context

that it refers to

edge connectivity

.

The (

vertex

)

connectivity

κ

⁢

(

G

)

is the minimum number of vertices (aka nodes) you have to remove to

either

make the graph no longer

connected

,

or

reduce it to a single vertex (node).

G

is said to be

k

-

(

vertex

)

-connected

for any

k

⩽

κ

⁢

(

G

)

∈

𝖨𝖭

. Note that “removing a vertex” in

graph theory

also involves removing all the edges

incident

to that vertex.

The

edge connectivity

κ

′

⁢

(

G

)

of a graph

G

is more straightforward, it is just the minimum number of edges you have to remove to make the graph no longer connected.

G

is said to be

k

-edge-connected

for any

k

⩽

κ

′

⁢

(

G

)

∈

𝖨𝖭

. And note “removing an edge” is simply that; it does not entail removing any vertices.

•

If

κ

′

⁢

(

G

)

=

0

also

κ

⁢

(

G

)

=

0

and vice versa; such graphs are
called disconnected and consist of several

connected components

.
If

κ

and

κ

′

are nonzero (the graph is 1-connected) it is
a

connected graph

(a single connected component)

•

If

κ

′

⁢

(

G

)

=

1

the graph contains at least one bridge. If

κ

′

⁢

(

G

)

>

1

(the graph is 2-edge-connected) every edge is
part of a cycle (

circuit

, closed path).

If

κ

⁢

(

G

)

=

1

the graph contains at least one

cutvertex

. If

κ

⁢

(

G

)

>

1

(the graph is 2-vertex-connected) there is, for
any pair of vertices, a cycle they both lie on.

•

For the

complete graphs

we have

κ

⁢

(

K

n

)

=

κ

′

⁢

(

K

n

)

=

n

-

1

.

•

For any graph

G

we have

κ

⁢

(

G

)

⩽

κ

′

⁢

(

G

)

⩽

δ

⁢

(

G

)

where the latter is the minimum

valency

in

G

(the valency of a
vertex is the number of edges sprouting from it).

Everything on this page applies equally well to

multigraphs

and

pseudographs

.

For

directed graphs

there are
two notions of connectivity (

http://planetmath.org/ConnectedGraph

) (“weak” if the underlying graph is connected, “strong” if you can get

from

everywhere

to

everywhere).

There are now
pictures (

http://planetmath.org/ExamplesOfKConnectedGraphs

) to go with this entry.

Title

k

-connected graph

Canonical name

kconnectedGraph

Date of creation

2013-03-22 13:10:58

Last modified on

2013-03-22 13:10:58

Owner

marijke (8873)

Last modified by

marijke (8873)

Numerical id

7

Author

marijke (8873)

Entry type

Definition

Classification

msc 05C40

Related topic

ConnectedGraph

Related topic

ClosedPath

Defines

k

-connected

k

-vertex-connected

k

-edge-connected connectivity

Generated on Thu Feb 8 20:44:24 2018 by

LaTeXML
Bondy and Chvátal theorem

Bondy and Chvátal theorem

Bondy and Chvátal’s theorem.

Let

G

be a graph of order

n

≥

3

and suppose that

u

and

v

are distinct non

adjacent vertices

such that

deg

⁡

(

u

)

+

deg

⁡

(

v

)

≥

n

.

Then

G

is

Hamiltonian

if and only if

G

+

u

⁢

v

is Hamiltonian.

Title

Bondy and Chvátal theorem

Canonical name

BondyAndChvatalTheorem

Date of creation

2013-03-22 11:52:57

Last modified on

2013-03-22 11:52:57

Owner

drini (3)

Last modified by

drini (3)

Numerical id

9

Author

drini (3)

Entry type

Theorem

Classification

msc 05C45

Classification

msc 81P99

Classification

msc 81S30

Classification

msc 81S99

Classification

msc 81-00

Classification

msc 81S05

Classification

msc 81P15

Related topic

HamiltonianGraph

Related topic

OresTheorem

Generated on Thu Feb 8 20:44:26 2018 by

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Euler circuit

Euler circuit

An

Euler circuit

is a

connected graph

such that starting at a vertex

a

, one can traverse along every edge of the graph once to each of the other vertices and return to vertex

a

. In other words, an Euler circuit is an

Euler path

that is a

circuit

. Thus, using the

properties

of odd and even

http://planetmath.org/node/788

degree vertices given in the

definition

of an Euler path, an Euler circuit exists if and only if every vertex of the graph has an even degree.

This graph is an Euler circuit as all vertices have degree 2.

This graph is not an Euler circuit.

Title

Euler circuit

Canonical

name

EulerCircuit

Date of creation

2013-03-22 12:02:06

Last modified on

2013-03-22 12:02:06

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Definition

Classification

msc 05C45

Synonym

Euler cycle

Related topic

EulerPath

Related topic

FleurysAlgorithm

Generated on Thu Feb 8 20:44:27 2018 by

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Fleury’s algorithm

Fleury’s algorithm

Fleury’s

algorithm

constructs an

Euler circuit

in a graph (if it’s possible).

1.

Pick any vertex to start

2.

From that vertex pick an edge to traverse, considering following rule: never cross a bridge of the

reduced

graph unless there is no other choice

3.

Darken that edge, as a reminder that you can’t traverse it again

4.

Travel that edge, coming to the next vertex

5.

Repeat 2-4 until all edges have been traversed, and you are back at the starting vertex

By “reduced graph” we mean the original graph minus the darkened (already used) edges.

Title

Fleury’s algorithm

Canonical name

FleurysAlgorithm

Date of creation

2013-03-22 13:35:15

Last modified on

2013-03-22 13:35:15

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Algorithm

Classification

msc 05C45

Related topic

EulerCircuit

Generated on Thu Feb 8 20:44:29 2018 by

LaTeXML
Hamiltonian cycle

Hamiltonian cycle

Let

G

be a graph. If there is a cycle visiting all vertices of

G

exactly once, we say that the cycle is a

Hamiltonian cycle

. A graph having a Hamiltonian cycle is called a

Hamiltonian graph

.

Title

Hamiltonian cycle

Canonical name

HamiltonianCycle

Date of creation

2013-03-22 11:52:49

Last modified on

2013-03-22 11:52:49

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 05C45

Classification

msc 81T70

Classification

msc 81T05

Classification

msc 81T25

Classification

msc 81T20

Classification

msc 81T75

Related topic

HamiltonianGraph

Related topic

HamiltonianPath

Related topic

GraphTheory

Related topic

Traceable

Generated on Thu Feb 8 20:44:31 2018 by

LaTeXML
Hamiltonian graph

Hamiltonian graph

A graph

G

is

Hamiltonian

if it has a

Hamiltonian cycle

.

A useful condition both necessary and

sufficient

for a graph to be Hamiltonian is not known. Ore’s

theorem

and the Bondy and Chvátal theorem give sufficient conditions, while a necessary condition follows quickly from the definition, namely:

Let

G

=

(

V

,

E

)

be a graph of order at least 3. If

G

is Hamiltonian, then for every

proper subset

U

of

V

, the

subgraph

induced by

V

-

U

has at most

|

U

|

components

.

Title

Hamiltonian graph

Canonical name

HamiltonianGraph

Date of creation

2013-03-22 11:52:43

Last modified on

2013-03-22 11:52:43

Owner

drini (3)

Last modified by

drini (3)

Numerical id

13

Author

drini (3)

Entry type

Definition

Classification

msc 05C45

Classification

msc 55-00

Classification

msc 82-00

Classification

msc 83-00

Classification

msc 81-00

Synonym

Hamiltonian

Related topic

HamiltonianCycle

Related topic

HamiltonianPath

Related topic

OresTheorem

Related topic

BondyAndChvatalTheorem

Related topic

PetersensGraph

Related topic

Traceable

Generated on Thu Feb 8 20:44:32 2018 by

LaTeXML
Hamiltonian path

Hamiltonian path

Let

G

be a graph. A path on

G

that includes every vertex exactly once is called a

Hamiltonian path

.

Title

Hamiltonian path

Canonical name

HamiltonianPath

Date of creation

2013-03-22 11:52:46

Last modified on

2013-03-22 11:52:46

Owner

drini (3)

Last modified by

drini (3)

Numerical id

10

Author

drini (3)

Entry type

Definition

Classification

msc 05C45

Classification

msc 46L05

Classification

msc 82-00

Classification

msc 83-00

Classification

msc 81-00

Related topic

HamiltonianCycle

Related topic

HamiltonianGraph

Related topic

PetersensGraph

Related topic

Traceable

Generated on Thu Feb 8 20:44:34 2018 by

LaTeXML
hypohamiltonian

hypohamiltonian

A graph

G

is

hypohamiltonian

if

G

is not

Hamiltonian

, but

G

-

v

is Hamiltonian for each

v

∈

V

(

V

the

vertex set

of

G

). The smallest hypohamiltonian graph is the

Petersen graph

, which has ten vertices.

Title

hypohamiltonian

Canonical name

Hypohamiltonian

Date of creation

2013-03-22 12:25:24

Last modified on

2013-03-22 12:25:24

Owner

digitalis (76)

Last modified by

digitalis (76)

Numerical id

4

Author

digitalis (76)

Entry type

Definition

Classification

msc 05C45

Classification

msc 05C65

Related topic

HamiltonianGraph

Related topic

PetersensGraph

Generated on Thu Feb 8 20:44:35 2018 by

LaTeXML
Ore’s theorem

Ore’s theorem

Let

G

be a

simple graph

of order

n

≥

3

such that, for every pair of distinct non

adjacent vertices

u

and

v

,

deg

⁡

(

u

)

+

deg

⁡

(

v

)

≥

n

.
Then

G

is a

Hamiltonian graph

.

Title

Ore’s theorem

Canonical name

OresTheorem

Date of creation

2013-03-22 11:52:40

Last modified on

2013-03-22 11:52:40

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

10

Author

Koro (127)

Entry type

Theorem

Classification

msc 05C45

Classification

msc 18D20

Related topic

HamiltonianGraph

Related topic

BondyAndChvatalTheorem

Generated on Thu Feb 8 20:44:37 2018 by

LaTeXML
Petersen graph

Petersen graph

Petersen’s graph

. An example of graph that is

traceable

but not

Hamiltonian

. That is, it has a

Hamiltonian path

but doesn’t have a

Hamiltonian cycle

.

This is also the

canonical

example of a hypohamiltonian graph.

Title

Petersen graph

Canonical name

PetersenGraph

Date of creation

2013-03-22 11:52:55

Last modified on

2013-03-22 11:52:55

Owner

drini (3)

Last modified by

drini (3)

Numerical id

10

Author

drini (3)

Entry type

Definition

Classification

msc 05C45

Classification

msc 46L05

Classification

msc 82-00

Classification

msc 83-00

Classification

msc 81-00

Related topic

Traceable

Related topic

HamiltonianPath

Related topic

HamiltonianGraph

Generated on Thu Feb 8 20:44:38 2018 by

LaTeXML
proof of Bondy and Chvátal theorem

proof of Bondy and Chvátal theorem

Proof.

The

sufficiency

of the condition is obvious and we shall prove the necessity by

contradiction

.

Assume that

G

+

u

⁢

v

is

Hamiltonian

but

G

is not.
Then

G

+

u

⁢

v

has a

Hamiltonian cycle

containing the edge

u

⁢

v

. Thus there exists a path

P

=

(

x

1

,

…

,

x

n

)

in

G

from

x

1

=

u

to

x

n

=

v

meeting all the vertices of

G

. If

x

i

is

adjacent

to

x

1

(

2

≤

i

≤

n

) then

x

i

-

1

is not adjacent to

x

n

, for otherwise

(

x

1

,

x

i

,

x

i

+

1

,

…

,

x

n

,

x

i

-

1

,

x

i

-

2

,

…

,

x

1

)

is a Hamiltonian cycle of

G

. Thus

d

⁢

(

x

n

)

≤

(

n

-

1

)

-

d

⁢

(

x

1

)

, that is

d

⁢

(

u

)

+

d

⁢

(

v

)

≤

n

-

1

, a contradiction
∎

Title

proof of Bondy and Chvátal

theorem

Canonical name

ProofOfBondyAndChvatalTheorem

Date of creation

2013-03-22 14:48:33

Last modified on

2013-03-22 14:48:33

Owner

taxipom (3607)

Last modified by

taxipom (3607)

Numerical id

8

Author

taxipom (3607)

Entry type

Proof

Classification

msc 05C45

Generated on Thu Feb 8 20:44:40 2018 by

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traceable

traceable

Let

G

be a graph. If

G

has a

Hamiltonian path

, we say that

G

is

traceable

.

Not every

traceable graph

is

Hamiltonian

. As an example consider Petersen’s graph.

Title

traceable

Canonical

name

Traceable

Date of creation

2013-03-22 11:52:52

Last modified on

2013-03-22 11:52:52

Owner

drini (3)

Last modified by

drini (3)

Numerical id

9

Author

drini (3)

Entry type

Definition

Classification

msc 05C45

Classification

msc 03B22

Classification

msc 03-00

Classification

msc 03-01

Related topic

HamiltonianPath

Related topic

HamiltonianCycle

Related topic

HamiltonianGraph

Related topic

PetersensGraph

Generated on Thu Feb 8 20:44:42 2018 by

LaTeXML
algebraic connectivity of a graph

algebraic connectivity of a graph

Let

L

⁢

(

G

)

be the

Laplacian matrix

(

http://planetmath.org/LaplacianMatrixOfAGraph

)
of a finite

connected graph

G

with

n

vertices. Let the eigenvalues
of

L

⁢

(

G

)

be denoted by

λ

1

≤

λ

2

≤

⋯

≤

λ

n

, which
is the usual notation in spectral graph theory.
The

connectivity

of

G

is

λ

2

.
The usual notation for the

algebraic connectivity

is

a

⁢

(

G

)

.
The

parameter

is a measure of how well the graph is connected.
For example,

a

⁢

(

G

)

≠

0

if and only if

G

is connected.

References

1

Fieldler, M. Algebraic connectivity of graphs,

Czech. Math. J.

23

(98) (1973)
pp. 298-305.

2

Merris, R. Laplacian matrices of graphs: a survey,

Lin.

Algebra

and its Appl.

197/198

(1994) pp. 143-176.

Title

algebraic connectivity of a graph

Canonical

name

AlgebraicConnectivityOfAGraph

Date of creation

2013-03-22 17:04:37

Last modified on

2013-03-22 17:04:37

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

9

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 05C50

Defines

algebraic connectivity

Generated on Thu Feb 8 20:44:54 2018 by

LaTeXML
Alon-Chung lemma

Alon-Chung lemma

Let

G

=

(

V

,

E

)

be a

undirected graph

of

n

vertices such that the degree of each vertex is equal to

d

. Let

X

be a subset of

V

. Then the number of edges in the

subgraph

induced by

X

is at most

1

2

⁢

n

⁢

(

d

⁢

|

X

|

2

+

λ

⁢

|

X

|

⁢

(

n

-

|

X

|

)

)

where

λ

is the second largest

eigenvalue

of the

adjacency matrix

of

G

.

Reference:

N. Alon and F. R. K. Chung, “Explicit construction of linear sized
tolerant

networks

,”

Discrete Math

., vol. 72, pp. 15-19, 1988.

Title

Alon-Chung lemma

Canonical name

AlonChungLemma

Date of creation

2013-03-22 17:26:47

Last modified on

2013-03-22 17:26:47

Owner

kshum (5987)

Last modified by

kshum (5987)

Numerical id

6

Author

kshum (5987)

Entry type

Theorem

Classification

msc 05C50

Generated on Thu Feb 8 20:44:55 2018 by

LaTeXML
incidence matrix with respect to an orientation

incidence matrix with respect to an orientation

Let

G

be a

finite graph

with

n

vertices,

{

v

1

,

…

,

v

n

}

and

m

edges,

{

e

1

,

…

,

e

m

}

.
For each edge

e

=

(

v

i

,

v

j

)

of

G

choose one vertex
to be the

positive

end and the other to be the negative end. In this way,
we assign an

orientation

to

G

. The

of

G

with respect an orientation is an

n

×

m

matrix

D

=

(

d

i

⁢

j

)

where

d

i

⁢

j

=

{

+

1

if

v

i

is the positive end of

e

j

-

1

if

v

i

is the negative end of

e

j

0

otherwise

.

Title

incidence matrix

with respect to an orientation

Canonical

name

IncidenceMatrixWithRespectToAnOrientation

Date of creation

2013-05-16 21:09:13

Last modified on

2013-05-16 21:09:13

Owner

Mathprof (13753)

Last modified by

unlord (1)

Numerical id

7

Author

Mathprof (1)

Entry type

Definition

Classification

msc 05C50

Defines

orientation

Generated on Thu Feb 8 20:44:57 2018 by

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Laplacian matrix of a graph

Laplacian matrix of a graph

Let

G

be a

finite graph

with

n

vectices and let

D

be the

incidence matrix

(

http://planetmath.org/
IncidenceMatrixWithRespectToAnOrientation

) of

G

with respect to some

orientation

.
The

Laplacian matrix

of

G

is defined to be

D

⁢

D

T

.

If we let

A

be the

adjacency matrix

of

G

then it can be shown
that

D

⁢

D

T

=

Δ

-

A

,
where

Δ

=

diag

⁢

(

δ

1

,

…

,

δ

n

)

and

δ

i

is the degree of
the vertex

v

i

. As a result, the Laplacian matrix is

independent

of what orientation is
chosen for

G

.

The Laplacian matrix is usually denoted by

L

⁢

(

G

)

. It is a

positive semidefinite

singular matrix

, so that the smallest

eigenvalue

is 0.

Title

Laplacian matrix of a graph

Canonical name

LaplacianMatrixOfAGraph

Date of creation

2013-03-22 17:04:40

Last modified on

2013-03-22 17:04:40

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 05C50

Defines

Laplacian matrix

Generated on Thu Feb 8 20:44:59 2018 by

LaTeXML
proof of Alon-Chung lemma

proof of Alon-Chung lemma

Let the vertices of

G

be labeled by

{

1

,

2

,

…

,

n

}

, and

𝐱

be the

column vector

defined by

x

i

=

{

1

if vertex

i

is in

X

0

otherwise

for

i

=

1

,

2

,

…

,

n

.

Let

𝐀

denote the

adjacency matrix

of

G

. The number of edges in the

subgraph

induced by

X

equals

1

2

⁢

𝐱

T

⁢

𝐀𝐱

, and we are going to show the following

equivalent

inequality

,

𝐱

T

⁢

𝐀𝐱

≤

1

n

⁢

(

d

⁢

|

X

|

2

+

λ

⁢

|

X

|

⁢

(

n

-

|

X

|

)

)

.

We label the

eigenvalues

of

𝐀

in decreasing order as

λ

1

≥

λ

2

≥

…

≥

λ

n

.

The largest eigenvalue

λ

1

is equal to the degree

d

, and we let

𝐮

1

be the corresponding normalized

eigenvector

,

𝐮

1

:=

1

n

⁢

[

1

,

1

,

…

,

1

]

T

.

As

𝐀

is

symmetric

, there is a

unitary matrix

𝐔

that

diagonalizes

𝐀

,

𝐔

T

⁢

𝐀𝐔

=

[

λ

1

λ

2

⋱

λ

n

]

.

The first column of

𝐔

is the column vector

𝐮

1

. We obtain

𝐱

T

⁢

𝐀𝐱

=

∑

k

=

1

n

λ

k

⁢

(

𝐮

k

T

⁢

𝐱

)

2

≤

d

⁢

(

𝐮

1

T

⁢

𝐱

)

2

+

λ

2

⁢

∑

k

=

2

n

(

𝐮

k

T

⁢

𝐱

)

2

.

In the line above, the first term is

d

⁢

(

𝐮

1

T

⁢

𝐱

)

2

=

d

⁢

|

X

|

2

n

,

while the summation is equal to

∑

k

=

2

n

(

𝐮

k

T

⁢

𝐱

)

2

=

∥

𝐱

∥

2

-

(

𝐮

1

T

⁢

𝐱

)

2

=

|

X

|

-

|

X

|

2

n

.

Hence

𝐱

T

⁢

𝐀𝐱

≤

d

⁢

|

X

|

2

n

+

λ

2

⁢

(

|

X

|

-

|

X

|

2

n

)

=

1

n

⁢

(

d

⁢

|

X

|

2

+

λ

2

⁢

|

X

|

⁢

(

n

-

|

X

|

)

)

.

Title

proof of Alon-Chung lemma

Canonical name

ProofOfAlonChungLemma

Date of creation

2013-03-22 17:26:53

Last modified on

2013-03-22 17:26:53

Owner

kshum (5987)

Last modified by

kshum (5987)

Numerical id

6

Author

kshum (5987)

Entry type

Proof

Classification

msc 05C50

Generated on Thu Feb 8 20:45:02 2018 by

LaTeXML
graph isomorphism

graph isomorphism

A

graph isomorphism

is a

bijection

between the vertices of two graphs

G

and

H

:

f

:

V

⁢

(

G

)

→

V

⁢

(

H

)

with the property that any two vertices

u

and

v

from

G

are

adjacent

if and only if

f

⁢

(

u

)

and

f

⁢

(

v

)

are adjacent in

H

.

If an

isomorphism

can be constructed between two graphs, then we say those graphs are

isomorphic

.

For example, consider these two graphs:

\xymatrix

⁢

a

⁢

\ar

⁢

@

-

[

r

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

⁢

d

]

⁢

&

⁢

g

⁢

b

⁢

\ar

⁢

@

-

[

r

⁢

u

]

⁢

\ar

⁢

@

-

[

r

]

⁢

\ar

⁢

@

-

[

r

⁢

d

⁢

d

]

⁢

&

⁢

h

⁢

c

⁢

\ar

⁢

@

-

[

r

⁢

u

⁢

u

]

⁢

\ar

⁢

@

-

[

r

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

i

⁢

d

⁢

\ar

⁢

@

-

[

r

⁢

u

⁢

u

]

⁢

\ar

⁢

@

-

[

r

⁢

u

]

⁢

\ar

⁢

@

-

[

r

]

⁢

&

⁢

j

\xymatrix

⁢

1

⁢

\ar

⁢

@

-

[

r

⁢

r

⁢

r

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

\ar

⁢

@

-

[

d

⁢

d

⁢

d

]

⁢

&

⁢

&

⁢

&

⁢

2

⁢

&

⁢

5

⁢

&

⁢

6

⁢

\ar

⁢

@

-

[

u

⁢

r

]

⁢

\ar

⁢

@

-

[

l

]

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

&

⁢

8

⁢

\ar

⁢

@

-

[

u

]

⁢

\ar

⁢

@

-

[

r

]

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

7

⁢

&

⁢

4

⁢

&

⁢

&

⁢

&

⁢

3

⁢

\ar

⁢

@

-

[

u

⁢

u

⁢

u

]

⁢

\ar

⁢

@

-

[

l

⁢

l

⁢

l

]

⁢

\ar

⁢

@

-

[

u

⁢

l

]

Although these graphs look very different at first, they are in fact isomorphic; one isomorphism between them is

f

⁢

(

a

)

=

1

f

⁢

(

b

)

=

6

f

⁢

(

c

)

=

8

f

⁢

(

d

)

=

3

f

⁢

(

g

)

=

5

f

⁢

(

h

)

=

2

f

⁢

(

i

)

=

4

f

⁢

(

j

)

=

7

Title

graph isomorphism

Canonical name

GraphIsomorphism

Date of creation

2013-03-22 12:16:24

Last modified on

2013-03-22 12:16:24

Owner

vampyr (22)

Last modified by

vampyr (22)

Numerical id

5

Author

vampyr (22)

Entry type

Definition

Classification

msc 05C60

Synonym

isomorphism

Synonym

isomorphic

Related topic

Graph

Related topic

Digraph

Related topic

GraphHomomorphism

Generated on Fri Feb 9 07:55:06 2018 by

LaTeXML
clique

clique

A maximal

http://planetmath.org/node/1757

complete

subgraph

of a graph is a

clique

, and the

clique number

ω

⁢

(

G

)

of a graph

G

is the

\PMlinkescapephrase

maximal order

maximal order of a clique in

G

. Simply,

ω

⁢

(

G

)

is the maximal order of a subgraph of

G

. Some authors however define a clique as any subgraph of

G

and refer to the other definition as

maximum clique

.

Adapted with permission of the author from

by Béla Bollobás, published by Springer-Verlag New York, Inc., 1998.

Title

clique

Canonical name

Clique

Date of creation

2013-03-22 12:30:53

Last modified on

2013-03-22 12:30:53

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

13

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 05C69

Related topic

IndependentSetAndIndependenceNumber

Defines

clique number

Defines

maximum clique

Generated on Fri Feb 9 07:55:25 2018 by

LaTeXML
independent set and independence number

independent set and independence number

A set of vertices in a graph

G

is called an

independent set

if there are no edges between the vertices.

The

independence number

of a graph

G

, usually denoted by

α

⁢

(

G

)

, is the size of a maximal independent set in

G

.

α

⁢

(

G

)

≥

ν

means that there are

ν

vertices with no edges between them.

An independent set is sometimes called a stable set or an anticlique.

Title

independent set and independence number

Canonical

name

IndependentSetAndIndependenceNumber

Date of creation

2013-03-22 14:30:06

Last modified on

2013-03-22 14:30:06

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 05C69

Synonym

stable set

Synonym

anticlique

Related topic

Clique2

Defines

independent set

Defines

independence number

Generated on Fri Feb 9 07:55:27 2018 by

LaTeXML
Mantel’s theorem

Mantel’s theorem

Every graph of order

n

and size greater than

⌊

n

2

/

4

⌋

contains a triangle (cycle of order 3).

Title

Mantel’s theorem

Canonical name

MantelsTheorem

Date of creation

2013-03-22 12:31:30

Last modified on

2013-03-22 12:31:30

Owner

digitalis (76)

Last modified by

digitalis (76)

Numerical id

5

Author

digitalis (76)

Entry type

Theorem

Classification

msc 05C69

Classification

msc 05C75

Related topic

Graph

Related topic

Cycle

Related topic

OrderOfAGraph

Related topic

SizeOfAGraph

Generated on Fri Feb 9 07:55:29 2018 by

LaTeXML
size of maximal independent set and chromatic number

size of maximal independent set and chromatic number

Let

α

⁢

(

G

)

be the size of the largest

independent set

in a graph

G

, and

χ

⁢

(

G

)

the

chromatic number

of

G

.

Theorem

:

α

⁢

(

G

)

⁢

χ

⁢

(

G

)

≥

|

G

|

.

Proof.

The vertices of

G

can be partitioned into

χ

⁢

(

G

)

monochromatic classes. Each class is an independent set, and hence cannot have size larger than

α

⁢

(

G

)

.
∎

Title

size of maximal independent set and chromatic number

Canonical name

SizeOfMaximalIndependentSetAndChromaticNumber

Date of creation

2013-03-22 14:30:02

Last modified on

2013-03-22 14:30:02

Owner

kshum (5987)

Last modified by

kshum (5987)

Numerical id

10

Author

kshum (5987)

Entry type

Theorem

Classification

msc 05C69

Classification

msc 05C15

Generated on Fri Feb 9 07:55:34 2018 by

LaTeXML