aarjaneiro/planetmath huggingface dataset:
complete bipartite graph

complete bipartite graph

The

complete bipartite graph

K

n

,

m

is a graph with two sets of vertices, one with

n

members and one with

m

, such that each vertex in one set is

adjacent

to every vertex in the other set and to no vertex in its own set. As the name implies,

K

n

,

m

is

bipartite

.

Examples of complete bipartite graphs:

K

2

,

5

:

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G

K

3

,

3

:

\xymatrix

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Title

complete bipartite graph

Canonical

name

CompleteBipartiteGraph

Date of creation

2013-03-22 12:17:13

Last modified on

2013-03-22 12:17:13

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

7

Author

yark (2760)

Entry type

Definition

Classification

msc 05C15

Generated on Thu Feb 8 20:42:25 2018 by

LaTeXML
complete k-partite graph

complete k-partite graph

A

complete k-partite graph

is a maximal

k

-partite
graph. In other words, a graph is

complete

k

-partite provided
that its

vertex set

admits a

partition

V

=

V

1

⊔

…

⊔

V

k

such that for any

u

∈

V

i

and

v

∈

V

j

,

uv

is an edge if and
only if

i

≠

j

.

For any tuple

(

a

1

,

…

,

a

k

)

, there is up to

graph isomorphism

a
single complete

k

-partite graph with maximal stable sets

V

1

,

…

,

V

k

such that for each

i

, there are exactly

a

i

vertices in

V

i

. This graph is denoted by

K

a

1

,

…

,

a

k

.

Below we display the 3-partite

complete graph

K

2

,

3

,

4

:

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&

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-

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]

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&

Title

complete k-partite graph

Canonical name

CompleteKpartiteGraph

Date of creation

2013-03-22 12:17:18

Last modified on

2013-03-22 12:17:18

Owner

mps (409)

Last modified by

mps (409)

Numerical id

8

Author

mps (409)

Entry type

Definition

Classification

msc 05C15

Synonym

complete

k

-partite graph

Generated on Thu Feb 8 20:42:27 2018 by

LaTeXML
Heawood number

Heawood number

The

Heawood number

of a

surface

is an upper bound for the

maximal
number

of colors needed to color any graph embedded in the
surface. In 1890 Heawood proved for all surfaces except the sphere
that no more than

H

⁢

(

S

)

=

⌊

7

+

49

-

24

⁢

e

⁢

(

S

)

2

⌋

colors are needed to color any graph embedded in a surface of

Euler characteristic

e

⁢

(

S

)

. The case of the sphere is the

four-color conjecture

which was settled by Appel and Haken in 1976. The number

H

⁢

(

S

)

became
known as Heawood number in 1976. Franklin proved that the

chromatic number

of a graph embedded in the

Klein bottle

can be as large as

6

,
but never exceeds

6

. Later it was proved in the works of Ringel
and Youngs that the

complete graph

of

H

⁢

(

S

)

vertices can be
embedded in the surface

S

unless

S

is the Klein bottle. This
established that Heawood’s bound could not be improved.

For example, the complete graph on

7

vertices can be embedded in the torus as follows:

{xy}

*

!

C

\xybox

\xymatrix

@

R

=

0

p

t

1

\ar

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-

[

r

]

\ar

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d

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\ar

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d

d

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&

2

\ar

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d

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&

3

\ar

@

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r

]

\ar

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d

d

r

]

&

1

\ar

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[

d

d

]

4

\ar

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d

d

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\ar

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d

r

]

&

&

&

4

\ar

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d

d

]

&

6

\ar

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r

]

\ar

@

-

[

d

d

d

r

]

&

7

\ar

@

-

[

d

d

d

]

\ar

@

-

[

d

d

d

r

]

\ar

@

-

[

d

r

]

\ar

@

-

[

u

r

]

&

5

\ar

@

-

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d

d

]

\ar

@

-

[

u

r

]

\ar

@

-

[

d

d

r

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\ar

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d

d

r

r

]

&

&

&

5

\ar

@

-

[

d

d

]

1

\ar

@

-

[

r

]

&

2

\ar

@

-

[

r

]

&

3

\ar

@

-

[

r

]

&

1

References

1

Béla Bollobás.

Graph Theory

: An Introductory Course

, volume 63 of

GTM

.

Springer-Verlag, 1979.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0411.05032

Zbl 0411.05032.

2

Thomas L. Saaty and Paul C. Kainen.

The

Four-Color Problem

: Assaults and Conquest

.

Dover, 1986.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0463.05041

Zbl 0463.05041.

Title

Heawood number

Canonical name

HeawoodNumber

Date of creation

2013-03-22 13:21:21

Last modified on

2013-03-22 13:21:21

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

12

Author

bbukh (348)

Entry type

Theorem

Classification

msc 05C15

Classification

msc 05C10

Related topic

FourColorConjecture

Generated on Thu Feb 8 20:42:30 2018 by

LaTeXML
k-partite graph

k-partite graph

A

k

-partite graph is a graph with a

chromatic number

of

k

.

An alternate definition of a

k

-partite graph is a graph where the vertices are partitioned into

k

subsets with the following conditions:

1.

No two vertices in the same subset are

adjacent

.

2.

There is no

partition

of the vertices with fewer than

k

subsets where condition 1 holds.

These two definitions are

equivalent

. Informally, we see that a colour can be assigned to all the vertices in each subset, since they are not adjacent to one another. Furthermore, this is also an optimal

colouring

, since the second condition holds.

An example of a 4-partite graph:

Generated on Thu Feb 8 20:42:37 2018 by

LaTeXML
property B

property B

A

hypergraph

G

is said to possess

property B

if it

2

-colorable, i.e., its vertices can be colored in two colors, so that no edge of

G

is monochromatic.

The

property

was named after Felix Bernstein by E. W. Miller.

Title

property B

Canonical

name

PropertyB

Date of creation

2013-03-22 13:39:10

Last modified on

2013-03-22 13:39:10

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

6

Author

bbukh (348)

Entry type

Definition

Classification

msc 05C15

Generated on Thu Feb 8 20:42:43 2018 by

LaTeXML
Tait coloring

Tait coloring

A

Tait coloring

of a trivalent (

http://planetmath.org/

Valency

) (aka cubic) graph is a

coloring

of its edges with only three colors, such that at each vertex the colors of the three edges there are different. After Peter G. Tait (1831–1901), Scottish physicist, who in 1880 proved the following

Theorem

(Tait)

A bridge-free trivalent

plane graph

can be
face-colored with 4 colors if and only if it can be edge-colored with 3
colors.

This

http://planetmath.org/node/6925

introduction

to face- and edge-colorings of plane graphs
has a

proof of the theorem

.

To put this in a modern

context

,

Vizing’s theorem

says that graphs

G

fall into two classes: those that can be colored with

Δ

⁢

(

G

)

colors and
those that need

Δ

⁢

(

G

)

+

1

colors, where

Δ

⁢

(

G

)

is the largest valency
that

occurs in

G

. When applied to trivalent graphs that means 3 and 4
colors, respectively. It is known that “almost all” are in the first class
— this

expression

has a technical meaning (for increasing graph size, the
proportion of them in the second class tends to zero).

Thanks to Tait’s result, it is a corollary of the

four-color theorem

that all planar trivalent graphs are in the first class (can be edge-3-colored).
The

converse

is not the case. Many trivalent graphs, in fact almost all of
them, can be edge-3-colored and yet are not planar.

K

3

,

3

is one example.

The

Petersen graph

is an example of a trivalent graph that needs 4 colors.

Title

Tait coloring

Canonical

name

TaitColoring

Date of creation

2013-03-22 15:10:29

Last modified on

2013-03-22 15:10:29

Owner

marijke (8873)

Last modified by

marijke (8873)

Numerical id

9

Author

marijke (8873)

Entry type

Definition

Classification

msc 05C15

Related topic

ColoringsOfPlaneGraphs

Related topic

VizingsTheorem

Generated on Thu Feb 8 20:42:45 2018 by

LaTeXML
cut

cut

On a

digraph

, define a

sink

to be a vertex with out-degree zero and a

source

to be a vertex with in-degree zero. Let

G

be a digraph with non-negative weights and with exactly one sink and exactly one source. A

cut

C

on

G

is a subset of the edges such that every path from the source to the sink

passes through

an edge in

C

. In other words, if we remove every edge in

C

from the graph, there is no longer a path from the source to the sink.

Define the weight of

C

as

W

C

=

∑

e

∈

C

W

⁢

(

e

)

where

W

⁢

(

e

)

is the weight of the edge

e

.

Observe that we may achieve a trivial cut by removing all the edges of

G

. Typically, we are more interested in

minimal cuts

, where the weight of the cut is minimized for a particular graph.

Title

cut

Canonical name

Cut

Date of creation

2013-03-22 13:00:53

Last modified on

2013-03-22 13:00:53

Owner

vampyr (22)

Last modified by

vampyr (22)

Numerical id

5

Author

vampyr (22)

Entry type

Definition

Classification

msc 05C20

Related topic

MaximumFlowminimumCutTheorem

Defines

minimum cut

Generated on Thu Feb 8 20:42:50 2018 by

LaTeXML
de Bruijn digraph

de Bruijn digraph

The vertices of the

de Bruijn digraph

B

⁢

(

n

,

m

)

are all possible words of length

m

-

1

chosen from an alphabet of size

n

.

B

⁢

(

n

,

m

)

has

n

m

edges consisting of each possible word of length

m

from an alphabet of size

n

. The edge

a

1

⁢

a

2

⁢

…

⁢

a

n

connects the vertex

a

1

⁢

a

2

⁢

…

⁢

a

n

-

1

to the vertex

a

2

⁢

a

3

⁢

…

⁢

a

n

.

For example,

B

⁢

(

2

,

4

)

could be drawn as:

Generated on Thu Feb 8 20:42:55 2018 by

LaTeXML
directed graph

directed graph

A

directed graph

or

digraph

is a pair

G

=

(

V

,

E

)

where

V

is a set of

vertices

and

E

is a subset of

V

×

V

called

edges

or

arcs

.

If

E

is

symmetric

(i.e.,

(

u

,

v

)

∈

E

if and only if

(

v

,

u

)

∈

E

), then the digraph is

isomorphic

to an ordinary (that is, undirected) graph.

Digraphs are generally drawn in a

similar

manner to graphs with arrows on the edges to indicate a sense of

direction

. For example, the digraph

(

{

a

,

b

,

c

,

d

}

,

{

(

a

,

b

)

,

(

b

,

d

)

,

(

b

,

c

)

,

(

c

,

b

)

,

(

c

,

c

)

,

(

c

,

d

)

}

)

may be drawn as

Generated on Thu Feb 8 20:43:01 2018 by

LaTeXML
flow

flow

On a

digraph

, define a

sink

to be a vertex with
out-degree zero and a

source

to be a vertex with
in-degree zero.
Let

G

be a digraph with non-negative weights and with exactly
one sink and exactly one source.
A

flow

on

G

is an assignment

f

:

E

⁢

(

G

)

→

ℝ

of values to each edge of

G

satisfying certain rules:

1.

For any edge

e

, we must have

0

≤

f

⁢

(

e

)

≤

W

⁢

(

e

)

(where

W

⁢

(

e

)

is the weight of

e

).

2.

For any vertex

v

, excluding the source and the sink, let

E

i

⁢

n

be
the set of edges

incident

to

v

and let

E

o

⁢

u

⁢

t

be the set of edges
incident from

v

.
Then we must have

∑

e

∈

E

i

⁢

n

f

⁢

(

e

)

=

∑

e

∈

E

o

⁢

u

⁢

t

f

⁢

(

e

)

.

Let

E

s

⁢

o

⁢

u

⁢

r

⁢

c

⁢

e

be the edges incident from the source, and let

E

s

⁢

i

⁢

n

⁢

k

be
the set of edges incident to the sink.
If

f

is a flow, then

∑

e

∈

E

s

⁢

i

⁢

n

⁢

k

f

⁢

(

e

)

=

∑

e

∈

E

s

⁢

o

⁢

u

⁢

r

⁢

c

⁢

e

f

⁢

(

e

)

.

We will refer to this quantity as the amount of flow.

Note that a flow given by

f

⁢

(

e

)

=

0

trivially

satisfies

these conditions.
We are typically more interested in

maximum flows

, where the amount
of flow is maximized for a particular graph.

We may interpret a flow as a means of transmitting something through
a

network

.
Suppose we think of the edges in a graph as pipes, with the weights
corresponding with the capacities of the pipes; we are pouring water
into the system through the source and draining it through the sink.
Then the first rule requires that we do not pump more water through
a pipe than is possible, and the second rule requires that any water
entering a junction of pipes must leave.
Under this

interpretation

, the maximum amount of flow corresponds to
the maximum amount of water we could pump through this network.

Instead of water in pipes, one may think of electric charge in a network of conductors.
Rule (2) above is one of Kirchoff’s two laws for such networks; the other
says that the sum of the voltage drops around any

circuit

is zero.

Title

flow

Canonical

name

Flow

Date of creation

2013-03-22 13:00:50

Last modified on

2013-03-22 13:00:50

Owner

bgins (4516)

Last modified by

bgins (4516)

Numerical id

6

Author

bgins (4516)

Entry type

Definition

Classification

msc 05C20

Classification

msc 94C15

Synonym

network flow

Related topic

MaximumFlowMinimumCutTheorem

Related topic

MaximumFlowminimumCutTheorem

Defines

maximum flow

Defines

source

Defines

sink

Defines

Kirchoff’s law

Generated on Thu Feb 8 20:43:03 2018 by

LaTeXML
Kautz graph

Kautz graph

The

Kautz graph

𝒦

K

N

+

1

is a

digraph

(

directed
graph

) of degree

K

and

dimension

N

+

1

, which has

(

K

+

1

)

⁢

K

N

vertices labeled by all possible strings

s

0

⁢

⋯

⁢

s

N

of length

N

+

1

which are composed of characters

s

i

chosen from an

alphabet

A

containing

K

+

1

distinct symbols, subject to the condition that

adjacent

characters in the string cannot be equal (

s

i

≠

s

i

+

1

).

The Kautz graph

𝒦

K

N

+

1

has

(

K

+

1

)

⁢

K

N

+

1

edges

{

(

s

0

⁢

s

1

⁢

⋯

⁢

s

N

,

s

1

⁢

s

2

⁢

⋯

⁢

s

N

⁢

s

N

+

1

)

|

s

i

∈

A

⁢

s

i

≠

s

i

+

1

}

.

(1)

It is natural to label each such edge of

𝒦

K

N

+

1

as

s

0

⁢

s

1

⁢

⋯

⁢

s

N

+

1

, giving a

one-to-one correspondence

between edges of the Kautz graph

𝒦

K

N

+

1

and vertices of the Kautz graph

𝒦

K

N

+

2

.

Example:

The Kautz graph

𝒦

2

2

has 6 nodes, and is depicted in the
following figure (using the alphabet

A

={0, 1, 2})

Properties

:

∙

The

diameter

of the Kautz graph

𝒦

K

N

is

N

(For example, there is a path of length

N

from

x

to

y

achieved by the

sequence

of edges

(

x

0

⁢

x

1

⁢

⋯

⁢

x

N

,

x

1

⁢

⋯

⁢

x

N

⁢

y

0

)

, …

(

x

N

⁢

y

0

⁢

⋯

⁢

y

N

-

1

,

y

0

⁢

⋯

⁢

y

N

)

unless

x

N

=

y

0

in
which case there is a

similar

path of length

N

-

1

beginning

(

x

0

⁢

x

1

⁢

⋯

⁢

x

N

-

1

⁢

y

0

,

x

1

⁢

⋯

⁢

x

N

-

1

⁢

y

0

⁢

y

1

)

,

…

.)

∙

Kautz graphs are closely related to

de Bruijn graphs

, which are
defined similarly but without the condition

s

i

≠

s

i

+

1

, and
with an alphabet of only

K

symbols for the degree

K

de Bruijn graph.

∙

For a fixed degree

K

and number of vertices

V

=

(

K

+

1

)

⁢

K

N

, the
Kautz graph has the smallest diameter of any possible directed graph
with

V

vertices and degree

K

.

∙

All Kautz graphs have

Eulerian cycles

(An Eulerian cycle
is one which visits each edge exactly once– This result follows
because Kautz graphs have in-degree equal to out-degree for each
node)

∙

All Kautz graphs have a

Hamiltonian cycle

(This result follows from the correspondence described above
between edges of the Kautz graph

𝒦

K

N

and vertices of the Kautz graph

𝒦

K

N

+

1

; a Hamiltonian cycle on

𝒦

K

N

+

1

is
given by an Eulerian cycle on

𝒦

K

N

)

∙

A degree-

k

Kautz graph has

k

disjoint

paths from any node

x

to any other node

y

.

Title

Kautz graph

Canonical

name

KautzGraph

Date of creation

2013-03-22 16:22:55

Last modified on

2013-03-22 16:22:55

Owner

wati (15191)

Last modified by

wati (15191)

Numerical id

7

Author

wati (15191)

Entry type

Definition

Classification

msc 05C20

Related topic

DeBruijnDigraph

Generated on Thu Feb 8 20:43:05 2018 by

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tournament

tournament

A

tournament

is a

directed graph

obtained by choosing a
direction for each edge in an undirected

complete graph

.
For example, here is a tournament on 4 vertices:

Generated on Thu Feb 8 20:43:15 2018 by

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Cayley graph

Cayley graph

Let

G

=

⟨

X

|

R

⟩

be a

presentation

of the

finitely generated group

G

with

generators

X

and

relations

R

. We define the

Cayley graph

Γ

=

Γ

⁢

(

G

,

X

)

of

G

with generators

X

as

Γ

=

(

G

,

E

)

,

where

E

=

{

{

u

,

a

⋅

u

}

|

u

∈

G

,

a

∈

X

}

.

That is, the vertices of the Cayley graph are precisely the elements of

G

, and two elements of

G

are

connected

by an edge iff some generator in

X

transfers the one to the other.

Examples

1.

G

=

ℤ

d

, with generators

X

=

{

e

1

,

…

,

e

d

}

, the standard basis vectors. Then

Γ

⁢

(

G

,

X

)

is the

d

-dimensional grid; confusingly, it too is often termed “

ℤ

d

”.

2.

G

=

F

d

, the

free group

with the

d

generators

X

=

{

g

1

,

…

,

g

d

}

. Then

Γ

⁢

(

G

,

X

)

is the

2

⁢

d

-

regular

tree.

A Cayley graph can be considered as a metric space with

d

⁢

(

x

,

y

)

(

x

,

y

∈

G

) being
the minimum number of edges one must traverse to get from

x

to

y

. Thus, each edge has length 1.

1) the graph is also edge labeled by the generator and directed. If you are in a group then given directed edges is sufficient to reconstruct the label.
Though in practice the vertices themselves are just dots (save the

identity

at times) in the graph and instead it is the edges that get labeled to tell you what to call the vertex.

2) The definition works for more than groups. In fact, it is sometimes used for the graph of a (semi)group acting on a set S as a Cayley graph with vertices S and edges given by the

group action

. As the definition of the graph is nearly identical authors do not generally strive to distinguish these two notions of a Cayley graph.

3) In a group, the graph is regular and connected.

4) The metric we mentioned above is usually called ”word length” . The bounding of word lengths – that is bounding the girth of a the graph, is a hard and active research problem involving the theory of expanders.

Title

Cayley graph

Canonical name

CayleyGraph

Date of creation

2013-03-22 12:48:53

Last modified on

2013-03-22 12:48:53

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 05C25

Related topic

Presentationgroup

Related topic

FinitelyGenerated

Generated on Thu Feb 8 20:43:18 2018 by

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hyperbolic group

hyperbolic group

A

finitely generated group

G

is

hyperbolic

if, for some

finite set

of

generators

A

of

G

, the

Cayley graph

Γ

⁢

(

G

,

A

)

, considered as a

metric space

with

d

⁢

(

x

,

y

)

being the minimum number of edges one must traverse to get from

x

to

y

, is a

hyperbolic metric space

.

Hyperbolicity

is a group-theoretic property. That is, if

A

and

B

are finite sets of generators of a group

G

and

Γ

⁢

(

G

,

A

)

is a hyperbolic metric space, then

Γ

⁢

(

G

,

B

)

is a hyperbolic metric space.

examples of hyperbolic groups include

finite groups

and

free groups

. If

G

is a finite group, then for any

x

,

y

∈

G

, we have that

d

⁢

(

x

,

y

)

≤

|

G

|

. (See the entry Cayley graph of

S

3

(

http://planetmath.org/CayleyGraphOfS_3

) for a pictorial example.) If

G

is a free group, then its Cayley graph is a

real tree

.

Title

hyperbolic group

Canonical name

HyperbolicGroup

Date of creation

2013-03-22 17:11:43

Last modified on

2013-03-22 17:11:43

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

6

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 05C25

Classification

msc 20F06

Classification

msc 54E35

Synonym

hyperbolicity

Related topic

RealTree

Generated on Thu Feb 8 20:43:21 2018 by

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enumerating graphs

enumerating graphs

How many graphs are there?

Proposition 1

.

The number of non-isomorphic

simple graphs

on

N

vertices, denoted

g

⁢

p

⁢

h

⁢

(

N

)

,

satisfies

2

(

N

2

)

N

!

≤

g

⁢

p

⁢

h

⁢

(

N

)

≤

2

(

N

2

)

.

Proof.

(

≤

)

Every simple graph on

N

vertices can be encoded by a

N

×

N

symmetric matrix

with all entries from

{

0

,

1

}

and with

0

’s on the

diagonal

. Therefore the information
can be encoded by strictly

upper triangular

N

×

N

matrices over

{

0

,

1

}

. This
gives

2

(

N

2

)

possibilities.

(

≥

)

The number of

isomorphisms

from a graph

G

to a graph

H

is equal to the number of

automorphisms

of

G

(or

H

). Therefore the total number of isomorphisms between two
graphs is no more than

N

!

– the total number of

permutations

of the

N

vertices of

G

. Therefore the total number of non-isomorphic simple graphs on

N

vertices is
at least

2

(

N

2

)

/

N

!

.

∎

•

The gap between the bounds is superexponential in

N

:
that is:

O

⁢

(

2

(

N

2

)

)

. However logarithmically the estimate is tight:

log

⁡

g

⁢

p

⁢

h

⁢

(

N

)

∈

Ω

⁢

(

N

2

-

N

⁢

log

⁡

N

)

∩

O

⁢

(

N

2

)

=

Θ

⁢

(

N

2

)

.

•

g

⁢

p

⁢

h

⁢

(

N

)

is closer to the lower bound; yet, most graphs have few automorphisms.

•

g

⁢

p

⁢

h

is monotonically increasing.

{xy}

(

0

,

10

)

*

Graphs on

N

vertices

,

(

0

,

0

)

*

[

o

]

=

<

40

p

t

>

G

*

\frm

o

,

(

5

,

0

)

*

∙

,

(

-

3

,

-

3

)

*

∙

,

(

-

4

,

2

)

*

∙

;

↪

{xy}

(

0

,

10

)

*

Graphs on

N

+

1

vertices

,

(

0

,

0

)

*

[

o

]

=

<

40

p

t

>

G

*

\frm

o

,

(

5

,

0

)

*

∙

,

(

-

3

,

-

3

)

*

∙

,

(

-

4

,

2

)

*

∙

,

(

10

,

0

)

*

∙

;

References

1

Harary, Frank and Palmer, Edgar M.

Graphical enumeration

Academic Press, New York, 1973, pp. xiv+271, 05C30, MR MR0357214 (50 #9682)

Title

enumerating graphs

Canonical name

EnumeratingGraphs

Date of creation

2013-03-22 15:50:07

Last modified on

2013-03-22 15:50:07

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

6

Author

Algeboy (12884)

Entry type

Theorem

Classification

msc 05C30

Related topic

EnumeratingGroups

Generated on Thu Feb 8 20:43:23 2018 by

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