aarjaneiro/planetmath huggingface dataset:
acyclic graph

acyclic graph

Any graph that contains no cycles is an

acyclic graph

. A

directed acyclic graph

is often called a DAG for short.

For example, the following graph and

digraph

are acyclic.

Generated on Thu Feb 8 20:43:45 2018 by

LaTeXML
bridges of Königsberg

bridges of Königsberg

The

bridges of Königsberg

is a famous problem inspired by an actual place and situation. The

solution

of the problem, put forth by Leonhard Euler in 1736, is widely considered to be the first work of

graph theory

and responsible for the foundation of the discipline.

The following figure shows a portion of the Prussian city of Königsberg. A river

passes through

the city, and there are two islands in the river. Seven bridges cross between the islands and the mainland:

Figure 1: Map of the Königsberg bridges.

The mathematical problem arose when citizens of Königsburg noticed that one could not take a stroll across all seven bridges, returning to the starting point, without crossing at least one bridge twice.

Answering the question of why this is the case required a mathematical theory that didn’t exist yet: graph theory. This was provided by Euler in his formative paper.

To solve the problem, we must

translate

it into a graph-theoretic

representation

. We model the land masses,

A

,

B

,

C

and

D

, as vertices in a graph. The bridges between the land masses become edges. This generates from the above picture the following graph:

Figure 2: Graph-theoretic representation of the Königsburg bridges.

At this point, we can apply what we know about

Euler paths

and

Euler circuits

. Since an Euler circuit for a graph exists only if every vertex has an even degree, the Königsberg graph must have no Euler circuit. Hence, we have explained why one cannot take a walk around Königsberg and return to the starting point without crossing at least one bridge more than once.

Title

bridges of Königsberg

Canonical name

BridgesOfKonigsberg

Date of creation

2013-03-22 12:33:47

Last modified on

2013-03-22 12:33:47

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

9

Author

akrowne (2)

Entry type

Topic

Classification

msc 05C38

Synonym

bridges of Koenigsberg

Generated on Thu Feb 8 20:43:48 2018 by

LaTeXML
cycle

cycle

A

cycle

in a graph,

digraph

, or

multigraph

, is a

simple path

from a vertex to itself (i.e., a path where the first vertex is the same as the last vertex and no edge is repeated).

For example, consider this graph:

\xymatrix

⁢

A

⁢

\ar

⁢

@

-

[

r

]

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

B

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

\ar

⁢

@

-

[

d

]

⁢

D

⁢

\ar

⁢

@

-

[

r

]

⁢

&

⁢

C

A

⁢

B

⁢

C

⁢

D

⁢

A

and

B

⁢

D

⁢

A

⁢

B

are two of the cycles in this graph.

A

⁢

B

⁢

A

is not a cycle, however, since it uses the edge connecting

A

and

B

twice.

A

⁢

B

⁢

C

⁢

D

is not a cycle because it begins on

A

but ends on

D

.

A cycle of length

n

is sometimes denoted

C

n

and may be referred to as a

polygon

of

n

sides: that is,

C

3

is a

triangle

,

C

4

is a

quadrilateral

,

C

5

is a

pentagon

, etc.

An

even

cycle is one of even length; similarly, an

odd

cycle is one of odd length.

Title

cycle

Canonical

name

Cycle

Date of creation

2013-03-22 12:17:21

Last modified on

2013-03-22 12:17:21

Owner

mps (409)

Last modified by

mps (409)

Numerical id

8

Author

mps (409)

Entry type

Definition

Classification

msc 05C38

Related topic

AcyclicGraph

Related topic

SimplePath

Related topic

VeblensTheorem

Related topic

MantelsTheorem

Related topic

Graph

Generated on Thu Feb 8 20:43:49 2018 by

LaTeXML
Euler path

Euler path

An

Euler path

in a graph is a path which traverses each edge of the graph exactly once. An Euler path which is a cycle is called an

Euler cycle

. For loopless graphs without isolated vertices, the existence of an Euler path implies the connectedness of the graph, since traversing every edge of such a graph requires visiting each vertex at least once.

If a

connected graph

has an Euler path, one can be constructed by applying Fleury’s

algorithm

. A connected graph has an Euler path if it has exactly zero or two vertices of odd degree. If every vertex has even degree, the graph has an Euler cycle.

This graph has an Euler cycle. All of its vertices are of even degree.

This graph has an Euler path which is not a cycle. It has exactly two vertices of odd degree.

Title

Euler path

Canonical name

EulerPath

Date of creation

2013-03-22 12:02:04

Last modified on

2013-03-22 12:02:04

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

18

Author

CWoo (3771)

Entry type

Definition

Classification

msc 05C38

Related topic

EulerCircuit

Related topic

Graph

Defines

Euler cycle

Generated on Thu Feb 8 20:43:51 2018 by

LaTeXML
girth

girth

The

girth

of a graph

G

is the length of the shortest cycle in

G

.

1

1

There is no widespread agreement on the girth of a forest, which has no cycles. It is also extremely unimportant.

For instance, the girth of any grid

ℤ

d

(where

d

>

2

) is 4, and the girth of the vertex graph of the

dodecahedron

is 5.

Title

girth

Canonical name

Girth

Date of creation

2013-03-22 12:45:56

Last modified on

2013-03-22 12:45:56

Owner

ariels (338)

Last modified by

ariels (338)

Numerical id

4

Author

ariels (338)

Entry type

Definition

Classification

msc 05C38

Related topic

ChromaticNumberAndGirth

Generated on Thu Feb 8 20:43:52 2018 by

LaTeXML
Laver table

Laver table

A

Laver table

L

n

for a given integer

n

>

0

has

2

n

rows

i

and columns

j

with each entry being determined thus:

L

n

⁢

(

i

,

j

)

=

i

⋆

j

, with

i

⋆

1

=

(

i

mod

2

n

)

+

1

for the first column. Subsequent rows are calculated with

i

⋆

(

j

⋆

k

)

:=

(

i

⋆

j

)

⋆

(

i

⋆

k

)

.

For example,

L

2

is

[

2

4

2

4

3

4

3

4

4

4

4

4

1

2

3

4

]

There is no known

closed formula

to calculate the entries of a Laver table directly, and it is in fact suspected that such a

formula

does not exist.

The entries repeat with a certain periodicity

m

. This periodicity is always a power of 2; the first few periodicities are 1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16, … (see A098820 in Sloane’s OEIS). The

sequence

is increasing, and it was proved in 1995 by Richard Laver that under the

assumption

that there exists a rank-into-rank, it actually tends towards

infinity

. Nevertheless, it grows extremely slowly; Randall Dougherty showed that the first

n

for which the table entries’ period can possibly be 32 is

A

⁢

(

9

,

A

⁢

(

8

,

A

⁢

(

8

,

255

)

)

)

, where

A

denotes the

Ackermann function

.

References

1

P. Dehornoy, ”Das Unendliche als Quelle der Erkenntnis”,

Spektrum der Wissenschaft Spezial

1/2001: 86 - 90

2

R. Laver, ”On the

Algebra

of

Elementary Embeddings

of a Rank into Itself”,

Advances in Mathematics

110

(1995): 334

This entry based entirely on a Wikipedia entry from a PlanetMath member.

Title

Laver table

Canonical

name

LaverTable

Date of creation

2013-03-22 16:26:13

Last modified on

2013-03-22 16:26:13

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 05C38

Generated on Thu Feb 8 20:43:54 2018 by

LaTeXML
Losanitsch’s triangle

Losanitsch’s triangle

A triangular

arrangement

of numbers very similar to Pascal’s

triangle

.

Begin as you would if you were constructing Pascal’s triangle, with a 1 in the top row, and that row

k

numbered 0, and the 1’s position

n

as 0.

1

1

1

1

x

1

⋮

⋮

⋮

Now, for the next value, add up the two values above, but then subtract

(

n

2

-

1

k

-

1

2

)

From this forward, do the same for every even-numbered position in an even-numbered row. Instead of calculating the

binomial coefficient

, it can be looked up in Pascal’s triangle.

1

1

1

1

1

1

1

2

2

1

1

2

4

2

1

1

3

6

6

3

1

1

3

9

10

9

3

1

1

4

12

19

19

12

4

1

⋮

⋮

⋮

This triangle was first studied by the Serbian chemist Sima Losanitsch, but has since been found to have applications in

graph theory

and

combinatorics

.

References

1

S. M. Losanitsch,

Die Isomerie-Arten bei den Homologen der Paraffin-Reihe

, Chem. Ber. 30 (1897), 1917-1926.

Title

Losanitsch’s triangle

Canonical

name

LosanitschsTriangle

Date of creation

2013-03-22 15:44:09

Last modified on

2013-03-22 15:44:09

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

13

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 05C38

Synonym

Lozanic’s triangle

Synonym

Lozanić’s triangle

Generated on Thu Feb 8 20:43:56 2018 by

LaTeXML
multiplicative encoding

multiplicative encoding

The

multiplicative encoding

of a

finite sequence

a

of

positive integers

k

long is the

product

of the first

k

primes with the members of the

sequence

used as exponents, thus

∏

i

=

1

k

p

i

a

i

,

with

p

i

being the

i

th

prime number

. For example, the fourth row of Pascal’s

triangle

is 1, 3, 3, 1. The multiplicative encoding is

2

1

⁢

3

3

⁢

5

3

⁢

7

1

=

47250

.

Encryption is not the purpose of multiplicative encoding, as the original sequence is easily retrieved with

trial division

. However, there are applications in

combinatorics

. Neil Sloane, for example, encodes the most famous number triangles as multiplicative encodings of the rows in order. While the resulting sequence for Pascal’s triangle does not consist of

squarefree

numbers (save the first two), it does contain only

singly even numbers

.

Another use is in logic, such as Kurt Gödel

encoding

a logical

proposition

as a single

integer

. As an example, Nagel and Newman convert

(

∃

x

)

(

x

=

s

y

)

to the

integer sequence

8, 4, 13, 9, 8, 13, 5, 7, 17, 9, and by multiplicative encoding to the single integer 172225505803959398742621651659678877886965404082311908389214945877004912002249920215937500000000.

References

1

Ernest Nagel & James Newman,

Gödel’s Proof

. New York: New York University Press (2001): 75 - 76

2

Neil Sloane,

The Encyclopedia of Integer Sequences

. New York: Academic Press (1995): M1722

Title

multiplicative encoding

Canonical name

MultiplicativeEncoding

Date of creation

2013-03-22 17:43:43

Last modified on

2013-03-22 17:43:43

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 05C38

Generated on Thu Feb 8 20:43:58 2018 by

LaTeXML
path

path

\xyoption

all

A

path

in a graph is a

finite sequence

of

alternating

vertices and edges, beginning and ending with a vertex,

v

1

⁢

e

1

⁢

v

2

⁢

e

2

⁢

v

3

⁢

…

⁢

e

n

-

1

⁢

v

n

such that every

consecutive

pair of vertices

v

x

and

v

x

+

1

are

adjacent

and

e

x

is

incident

with

v

x

and with

v

x

+

1

. Typically, the edges may be omitted when writing a path (e.g.,

v

1

⁢

v

2

⁢

v

3

⁢

…

⁢

v

n

) since only one edge of a graph may connect two

adjacent vertices

. In a

multigraph

, however, the choice of edge may be significant.

The length of a path is the number of edges in it.

Consider the following graph:

\xymatrix

⁢

A

⁢

\ar

⁢

@

-

[

r

]

⁢

&

⁢

B

⁢

\ar

⁢

@

-

[

d

]

⁢

D

⁢

\ar

⁢

@

-

[

u

]

⁢

&

⁢

C

⁢

\ar

⁢

@

-

[

l

]

Paths include (but are certainly not limited to)

A

⁢

B

⁢

C

⁢

D

(length 3),

A

⁢

B

⁢

C

⁢

D

⁢

A

(length 4), and

A

⁢

B

⁢

A

⁢

B

⁢

A

⁢

B

⁢

A

⁢

B

⁢

A

⁢

D

⁢

C

⁢

B

⁢

A

(length 12).

A

⁢

B

⁢

D

is not a path since

B

is not adjacent to

D

.

In a

digraph

, each consecutive pair of vertices must be

connected

by an edge with the proper

orientation

; if

e

=

(

u

,

v

)

is an edge, but

(

v

,

u

)

is not, then

u

⁢

e

⁢

v

is a valid path but

v

⁢

e

⁢

u

is not.

Consider this digraph:

\xymatrix

⁢

G

⁢

\ar

⁢

[

r

]

⁢

\ar

⁢

[

d

]

⁢

&

⁢

H

⁢

\ar

⁢

[

d

]

⁢

\ar

⁢

[

l

]

⁢

J

⁢

&

⁢

I

⁢

\ar

⁢

[

l

]

G

⁢

H

⁢

I

⁢

J

,

G

⁢

J

, and

G

⁢

H

⁢

G

⁢

H

⁢

G

⁢

H

are all valid paths.

G

⁢

H

⁢

J

is not a valid path because

H

and

J

are not connected.

G

⁢

J

⁢

I

is not a valid path because the edge connecting

I

to

J

has the opposite orientation.

Title

path

Canonical name

Path1

Date of creation

2013-03-22 12:16:49

Last modified on

2013-03-22 12:16:49

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

8

Author

CWoo (3771)

Entry type

Definition

Classification

msc 05C38

Related topic

ClosedPath

Defines

path length

Generated on Thu Feb 8 20:44:00 2018 by

LaTeXML
proof of Veblen’s theorem

proof of Veblen’s theorem

The proof is very easy by

induction

on the number of elements of
the set

E

of edges.
If

E

is empty, then all the vertices have degree zero, which is even.
Suppose

E

is nonempty.
If the graph contains no cycle, then some vertex has degree

1

, which is odd.
Finally, if the graph does contain a cycle

C

, then every vertex has
the same degree mod

2

with respect to

E

-

C

, as it has with respect
to

E

, and we can conclude by induction.

Title

proof of Veblen’s

theorem

Canonical name

ProofOfVeblensTheorem

Date of creation

2013-03-22 13:56:51

Last modified on

2013-03-22 13:56:51

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

4

Author

mathcam (2727)

Entry type

Proof

Classification

msc 05C38

Generated on Thu Feb 8 20:44:05 2018 by

LaTeXML
simple path

simple path

A

simple path

in a graph is a path

P

=

v

0

⁢

e

0

⁢

v

1

⁢

…

⁢

e

n

-

1

⁢

v

n

such that no vertex occurs twice in

P

.
Some authors relax this condition by permitting

v

0

=

v

n

. In this case the path is usually called a cycle.

Title

simple path

Canonical

name

SimplePath

Date of creation

2013-03-22 12:30:42

Last modified on

2013-03-22 12:30:42

Owner

mps (409)

Last modified by

mps (409)

Numerical id

6

Author

mps (409)

Entry type

Definition

Classification

msc 05C38

Related topic

Path

Related topic

Cycle

Related topic

Graph

Related topic

PathConnected

Generated on Thu Feb 8 20:44:06 2018 by

LaTeXML
Veblen’s theorem

Veblen’s theorem

The

edge set

of a graph can be partitioned (

http://planetmath.org/

Partition

) into cycles if and only if every vertex has even degree.

Title

Veblen’s

theorem

Canonical name

VeblensTheorem

Date of creation

2013-03-22 12:31:27

Last modified on

2013-03-22 12:31:27

Owner

digitalis (76)

Last modified by

digitalis (76)

Numerical id

5

Author

digitalis (76)

Entry type

Theorem

Classification

msc 05C38

Related topic

Graph

Related topic

Cycle

Related topic

Digraph

Related topic

DegreeOfAVertex

Related topic

DirectedGraph

Generated on Thu Feb 8 20:44:08 2018 by

LaTeXML
connected graph

connected graph

A

connected graph

is a graph such that there exists a path between all pairs of vertices. If the graph is a

directed graph

, and there exists a path from each vertex to every other vertex, then it is a

strongly connected graph

.

A

connected component

is a maximal (under inclusion) subset of vertices of any graph and any edges between them that forms a connected graph. Similarly, a

strongly connected component

is a maximal (under inclusion) subset of vertices of any

digraph

and any edges between them that forms a strongly connected graph. Any graph or digraph is a union of

connected

or strongly connected components, plus some edges to join the components together. Thus any graph can be decomposed into its connected or strongly connected components. For instance, Tarjan’s

algorithm

can decompose any digraph into its strongly connected components.

For example, the following graph and digraph are connected and strongly connected, respectively.

Generated on Thu Feb 8 20:44:14 2018 by

LaTeXML
cut-vertex

cut-vertex

A

cut-vertex

of a graph
is a vertex whose deletion increases the number of

components

of the graph.
The edge analogue of a cut-vertex is a bridge.

Title

cut-vertex

Canonical

name

Cutvertex

Date of creation

2013-03-22 12:31:38

Last modified on

2013-03-22 12:31:38

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

6

Author

yark (2760)

Entry type

Definition

Classification

msc 05C40

Synonym

cutvertex

Synonym

cut vertex

Related topic

Bridge

Related topic

ConnectedGraph

Related topic

Block

Generated on Thu Feb 8 20:44:15 2018 by

LaTeXML
examples of k-connected graphs

examples of

k

-connected graphs

Some examples to go with the

k

-

connected graph

(

http://planetmath.org/KConnectedGraph

) entry, where

κ

and

κ

′

are defined.

examples of disconnected graphs:

examples of bridges & cutvertices:

Legend:

b

is a bridge,

c

a

cutvertex

examples of better connected graphs:

Title

examples of

k

-connected graphs

Canonical name

ExamplesOfkconnectedGraphs

Date of creation

2013-03-22 15:10:14

Last modified on

2013-03-22 15:10:14

Owner

marijke (8873)

Last modified by

marijke (8873)

Numerical id

4

Author

marijke (8873)

Entry type

Example

Classification

msc 05C40

Generated on Thu Feb 8 20:44:17 2018 by

LaTeXML
Thomassen’s theorem on 3-connected graphs

Thomassen’s theorem on

3

-connected graphs

Every

3

-

connected

(

http://planetmath.org/KConnectedGraph

) graph

G

with more
than 4 vertices has an edge

e

such that

G

/

e

(

http://planetmath.org/EdgeContraction

) is also

3

-connected.

Note:

G

/

e

denotes the graph obtained from

G

by contracting the edge

e

.
If

e

=

x

⁢

y

we also use the notation

G

/

x

⁢

y

.

Suppose such an edge doesn’t exist. Then, for every edge

e

=

x

⁢

y

, the
graph

G

/

e

isn’t

3

-connected and can be made disconnected by
removing 2 vertices. Since

κ

⁢

(

G

)

≥

3

, our contracted vertex

v

x

⁢

y

has to be one of these two. So for every edge

e

,

G

has a
vertex

z

≠

x

,

y

such that

{

v

x

⁢

y

,

z

}

separates

G

/

e

. Any 2
vertices

separated

by

{

v

x

⁢

y

,

z

}

in

G

/

e

are separated in

G

by

S

:=

{

x

,

y

,

z

}

. Since the

minimal

size of a separating set is 3, every
vertex in

S

has an

adjacent vertex

in every component of

G

-

S

.

Now we choose the edge

e

, the vertex

z

and the component

C

such
that

|

C

|

is minimal. We also choose a vertex

v

adjacent

to

z

in

C

.

By construction

G

/

z

⁢

v

is not

3

-connected since removing

x

⁢

y

disconnects

C

-

v

from

G

/

z

⁢

v

. So there is a vertex

w

such that

{

z

,

v

,

w

}

separates

G

and as above every vertex in

{

z

,

v

,

w

}

has
an adjacent vertex in every component of

G

-

{

z

,

v

,

w

}

. We now
consider a component

D

of

G

-

{

z

,

v

,

w

}

that doesn’t contain

x

or

y

. Such a component exists since

x

and

y

belong to the same
component and

G

-

{

z

,

v

,

w

}

isn’t connected. Any vertex adjacent to

v

in

D

is also an element of

C

since

v

is an element of

C

. This means

D

is a

proper subset

of

C

which contradicts our

assumption

that

|

C

|

was minimal.

Title

Thomassen’s

theorem

on

3

-

connected graphs

Canonical name

ThomassensTheoremOn3connectedGraphs

Date of creation

2013-03-22 13:11:09

Last modified on

2013-03-22 13:11:09

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

6

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 05C40

Related topic

EdgeContraction

Generated on Thu Feb 8 20:44:19 2018 by

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Tutte’s wheel theorem

Tutte’s wheel theorem

Every

3

-

connected

(

http://planetmath.org/KConnectedGraph

)

simple graph

can be constructed starting from a

wheel graph

by repeatedly either adding an edge between two non-adjacent vertices or splitting a vertex.

Title

Tutte’s wheel theorem

Canonical name

TuttesWheelTheorem

Date of creation

2013-03-22 13:11:06

Last modified on

2013-03-22 13:11:06

Owner

lieven (1075)

Last modified by

lieven (1075)

Numerical id

4

Author

lieven (1075)

Entry type

Theorem

Classification

msc 05C40

Generated on Thu Feb 8 20:44:21 2018 by

LaTeXML
valency

valency

In a graph,

multigraph

, or

pseudograph

G

, the

valency

of a vertex is
the number of edges attached to it (note that a loop counts twice).

Synonymous with and
. There are some unrelated things also
called valence; there are of course many things all called degree.

For

directed graphs

,

in-

and

out-

are prefixed to any of the synonyms, to count incoming and outgoing edges separately.

If

ρ

⁢

(

v

)

is used for the valency of vertex

v

, the notation

ρ

⁢

(

G

)

(or

ρ

on its own if there is no scope for confusion) denotes the maximum valency found in graph

G

. Another notation often seen is

δ

⁢

(

G

)

and

Δ

⁢

(

G

)

for lowest and highest valency in

G

respectively.

If the valency is the same number (

ρ

, say) for all its vertices,

G

is called

regular

. More specifically it is called

ρ

-valent

or

ρ

-regular.

Connected

(components of)…

•

…0-valent graphs are edgeless vertices,

•

…1-valent graphs are pairs of vertices joined by an edge,

•

…2-valent graphs are cyclic graphs, i.e.

n

-gons, of various sizes

•

From

ρ

⩾

3

these structures start getting more interesting.
3-valent (or

trivalent

) graphs are also known as

cubic graphs

.

A

ρ

-valent graph with

n

vertices has

n

⁢

ρ

/

2

edges.

Title

valency

Canonical

name

Valency

Date of creation

2013-03-22 15:10:17

Last modified on

2013-03-22 15:10:17

Owner

marijke (8873)

Last modified by

marijke (8873)

Numerical id

6

Author

marijke (8873)

Entry type

Definition

Classification

msc 05C40

Synonym

valence

Synonym

degree

Defines

ρ

-valent

Defines

trivalent graph

Defines

cubic graph

Defines

regular

Defines

regular graph

Generated on Thu Feb 8 20:44:22 2018 by

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