aarjaneiro/planetmath huggingface dataset:
tree traversals

tree traversals

A

tree traversal

is an

algorithm

for visiting all the nodes (

http://planetmath.org/Graph

) in a

rooted tree

exactly once. The constraint is on rooted trees, because the root is taken to be the starting point of the traversal. A traversal is also defined on a forest in the sense that each tree in the forest can be iteratively traversed (provided one knows the roots of every tree beforehand). This entry presents a few common and simple tree traversals.

In the description of a tree (

http://planetmath.org/forest

), the notion of rooted-subtrees
was presented. Full understanding of this notion is necessary to understand the traversals presented here, as each of these traversals depends heavily upon this notion.

In a traversal, there is the notion of

visiting

a node. Visiting a node often consists of doing some

computation

with that node. The traversals are defined here without any notion of what is being done to visit a node, and simply indicate where the visit occurs (and most importantly, in what order).

Examples of each traversal will be illustrated on the following

binary tree

.

Generated on Thu Feb 8 20:41:11 2018 by

LaTeXML
trie

trie

A

trie

is a digital tree for storing a set of strings in which there is one node for every prefix of every string in the set. The name comes from the word
re

trie

val, and thus is pronounced the same as

tree

(which leads to much confusion when spoken aloud). The word retrieval is stressed, because a trie has a lookup time that is

equivalent

to the length of the string being looked up.

If a trie is to store some set of strings

S

⊆

Σ

*

(where

Σ

is an

alphabet

),
then it takes the following form.
Each edge leading to non-leaf nodes in the trie is labelled by an element of

Σ

. Any edge leading to a leaf node is labelled by

$

(some symbol

not

in

Σ

). For every string

s

∈

S

, there is a path from the root of the trie to a leaf, the labels of which when concatenated form

s

+

+

$

(where

+

⁣

+

is the string concatenation

operator

). For every path from the root of the trie to a leaf, the labels of the edges concatenated form some string in

S

.

Example

Suppose we wish to store the set of strings

S

:=

{

a

⁢

l

⁢

p

⁢

h

⁢

a

,

b

⁢

e

⁢

t

⁢

a

,

b

⁢

e

⁢

a

⁢

r

,

b

⁢

e

⁢

a

⁢

s

⁢

t

,

b

⁢

e

⁢

a

⁢

t

}

. The trie that stores

S

would be

Generated on Thu Feb 8 20:41:16 2018 by

LaTeXML
weighted path length

weighted path length

Given an

extended binary tree

T

(that is, simply any

complete binary tree

, where leafs are denoted as

external nodes

), associate weights with each external node. The

weighted path length

of

T

is the sum of the

product

of the weight and

path length

of each external node, over all external nodes.

Another formulation is that weighted path length is

∑

w

j

⁢

l

j

over all external nodes

j

, where

w

j

is the weight of an external node

j

, and

l

j

is the

distance

from the root of the tree to

j

. If

w

j

=

1

for all

j

, then weighted path length is exactly the same as

external path length

.

Example

Let

T

be the following extended binary tree. Square nodes are external nodes, and circular nodes are

internal nodes

. Values in external nodes indicate weights, which are given in this problem, while values in internal nodes represent the weighted path length of

subtrees

rooted at those nodes, and are calculated from the given weights and the given tree. The weight of the tree as a whole is given at the root of the tree.

This tree happens to give the

minimum weighted path length

for this particular set of weights.

Title

weighted path length

Canonical

name

WeightedPathLength

Date of creation

2013-03-22 12:32:09

Last modified on

2013-03-22 12:32:09

Owner

Logan (6)

Last modified by

Logan (6)

Numerical id

4

Author

Logan (6)

Entry type

Definition

Classification

msc 05C05

Related topic

ExternalPathLength

Related topic

ExtendedBinaryTree

Related topic

CompleteBinaryTree

Related topic

MinimumWeightedPathLength

Generated on Thu Feb 8 20:41:21 2018 by

LaTeXML
crossing lemma

crossing lemma

The

crossing number

of a graph (

http://planetmath.org/Graph

)

G

with

n

vertices (

http://planetmath.org/Graph

) and

m

≥

4

⁢

n

edges (

http://planetmath.org/Graph

) is

cr

⁡

(

G

)

≥

1

64

⁢

m

3

n

2

.

Title

crossing lemma

Canonical name

CrossingLemma

Date of creation

2013-03-22 13:20:37

Last modified on

2013-03-22 13:20:37

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

6

Author

bbukh (348)

Entry type

Theorem

Classification

msc 05C10

Related topic

PlanarGraph

Generated on Thu Feb 8 20:41:23 2018 by

LaTeXML
crossing number

crossing number

The

crossing number

cr

⁡

(

G

)

of a graph (

http://planetmath.org/Graph

)

G

is the

minimal number

of crossings among all

embeddings

of

G

in the plane.

Title

crossing number

Canonical

name

CrossingNumber

Date of creation

2013-03-22 13:20:34

Last modified on

2013-03-22 13:20:34

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

5

Author

bbukh (348)

Entry type

Definition

Classification

msc 05C10

Related topic

PlanarGraph

Generated on Thu Feb 8 20:41:25 2018 by

LaTeXML
example of planar graph with two different embeddings into the plane

example of planar graph with two different embeddings into the plane

Consider the following two

plane graphs

:

The plane graph

G

1

.

The plane graph

G

2

.

Clearly the

multigraph

associated with each of these plane graphs is the same, but the two plane graphs are not

isomorphic

: in

G

1

, every face is

adjacent

to the

central

region, while in

G

2

there is no face adjacent to every other. This also implies that their

dual graphs

are non-isomorphic.

Title

example of

planar graph

with two different embeddings into the plane

Canonical name

ExampleOfPlanarGraphWithTwoDifferentEmbeddingsIntoThePlane

Date of creation

2013-03-22 14:17:03

Last modified on

2013-03-22 14:17:03

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

5

Author

archibal (4430)

Entry type

Example

Classification

msc 05C10

Generated on Thu Feb 8 20:41:26 2018 by

LaTeXML
four-color conjecture

four-color conjecture

The

four-color conjecture

was a long-standing problem posed by Francis Guthrie to his professor Augustus De Morgan in 1852, while

coloring

a map of England. The conjecture states that every map on a plane or a sphere can be colored using only four colors such that no two

adjacent

countries are assigned the same color. There are maps that need four colors to be colored as the figure below shows. This conjecture

equivalent

to the statement that

chromatic number

of every

planar graph

is no more than four.

Figure 1:

Four mutually
adjacent countries

After many unsuccessful attempts the conjecture was proven by Appel and Haken in 1976 with the aid of a computer. Before it was known that every planar map can be five-colored by the work of Heawood in 1890.

Interestingly, the seemingly harder problem of determining the

maximal number

of colors needed for all surfaces other than the
sphere was solved long before the four-color conjecture was
settled. This is the

Heawood number

of the surface unless the surface
is the

Klein bottle

in which case it is

6

.

References

1

Kenneth O. May.

The origin of the four-color conjecture.

Isis

, 56(3):346–348, 1965.

http://links.jstor.org/sici?sici=0021-1753%28196523%2956%3A3%3C346%3ATOOTFC%3E2.0.CO%3B2-Z

Available
online at

http://www.jstor.org

JSTOR.

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

four-color conjecture

Canonical name

FourcolorConjecture

Date of creation

2013-03-22 13:21:18

Last modified on

2013-03-22 13:21:18

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

19

Author

bbukh (348)

Entry type

Theorem

Classification

msc 05C10

Classification

msc 05C15

Synonym

Appel-Haken theorem

Synonym

4-color conjecture

Related topic

PlanarGraph

Related topic

ChromaticNumber

Related topic

HeawoodNumber

Generated on Thu Feb 8 20:41:29 2018 by

LaTeXML
Kuratowski’s theorem

Kuratowski’s theorem

A

finite graph

is planar if and only if it contains no

subgraph

that is

isomorphic

to or is a

subdivision

of

K

5

or

K

3

,

3

, where

K

5

is the

complete graph

of order 5 and

K

3

,

3

is the

complete bipartite graph

with 3 vertices in each of the halfs. Wagner’s

theorem

is an

equivalent

later result.

References

1

Kazimierz Kuratowski.

Sur le problème des courbes gauches en topologie.

Fund. Math.

, 15:271–283, 1930.

Title

Kuratowski’s theorem

Canonical name

KuratowskisTheorem

Date of creation

2013-03-22 11:57:45

Last modified on

2013-03-22 11:57:45

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

12

Author

bbukh (348)

Entry type

Theorem

Classification

msc 05C10

Related topic

PlanarGraph

Related topic

WagnersTheorem

Generated on Thu Feb 8 20:41:30 2018 by

LaTeXML
proof of crossing lemma

proof of crossing lemma

Euler’s formula implies the linear

lower bound

cr

⁡

(

G

)

≥

m

-

3

⁢

n

+

6

, and so it cannot be used directly. What we need is to consider the

subgraphs

of our graph, apply Euler’s formula on them, and then combine the estimates. The

probabilistic method

provides a natural way to do that.

Consider a minimal embedding of

G

. Choose independently every vertex of

G

with

probability

p

. Let

G

p

be a graph induced by those vertices. By Euler’s formula,

cr

⁡

(

G

p

)

-

m

p

+

3

⁢

n

p

≥

0

. The

expectation

is clearly

E

⁢

(

cr

⁡

(

G

p

)

-

m

p

+

3

⁢

n

p

)

≥

0

.

Since

E

⁢

(

n

p

)

=

p

⁢

n

,

E

⁢

(

m

p

)

=

p

2

⁢

m

and

E

⁢

(

X

p

)

=

p

4

⁢

cr

⁡

(

G

)

, we get an inequality that bounds the

crossing number

of

G

from below,

cr

⁡

(

G

)

≥

p

-

2

⁢

m

-

3

⁢

p

-

3

⁢

n

.

Now set

p

=

4

⁢

n

m

(which is at most

1

since

m

≥

4

⁢

n

), and the inequaliy becomes

cr

⁡

(

G

)

≥

1

64

⁢

m

3

n

2

.

Similarly, if

m

≥

9

2

⁢

n

, then we can set

p

=

9

⁢

n

2

⁢

m

to get

cr

⁡

(

G

)

≥

4

243

⁢

m

3

n

2

.

References

1

Martin Aigner and Günter M. Ziegler.

Proofs from THE BOOK

.

Springer, 1999.

Title

proof of crossing lemma

Canonical name

ProofOfCrossingLemma

Date of creation

2013-03-22 13:38:46

Last modified on

2013-03-22 13:38:46

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

6

Author

bbukh (348)

Entry type

Proof

Classification

msc 05C10

Classification

msc 05C80

Related topic

ProbabilisticMethod

Generated on Thu Feb 8 20:41:35 2018 by

LaTeXML
diameter

diameter

The

diameter

of a

connected graph

(or

multigraph

or

pseudograph

) is the largest

distance

that occurs in it.

Here

distance

, between two nodes (vertices), refers to the length of the shortest path between them.

In

disconnected graphs

the diameter is undefined (sometimes it is said to be infinite).

Title

diameter

Canonical

name

Diameter12

Date of creation

2013-03-22 15:11:24

Last modified on

2013-03-22 15:11:24

Owner

marijke (8873)

Last modified by

marijke (8873)

Numerical id

7

Author

marijke (8873)

Entry type

Definition

Classification

msc 05C12

Related topic

Graph

Related topic

ClosedPath

Related topic

DistanceInAGraph

Generated on Thu Feb 8 20:41:37 2018 by

LaTeXML
distance (in a graph)

distance (in a graph)

The

distance

d

⁢

(

x

,

y

)

of two vertices

x

and

y

of a graph

G

is the length of the shortest path (or, equivalently, walk) from

x

to

y

. If there is no path from

x

to

y

(i.e. if they lie in different

components

of G), we set

d

⁢

(

x

,

y

)

:=

∞

.

Two basic graph invariants involving distance are the

diameter

diam

⁡

G

:=

max

(

x

,

y

)

∈

V

⁢

(

G

)

2

⁡

d

⁢

(

x

,

y

)

(the maximum distance between two vertices of

G

) and the

radius

rad

⁡

G

:=

min

x

∈

V

⁢

(

G

)

⁡

max

y

∈

V

⁢

(

G

)

⁡

d

⁢

(

x

,

y

)

(the maximum distance of a vertex from a

central

vertex of

G

, i.e. a vertex such that the maximum distance to another vertex is minimal).

Title

distance (in a graph)

Canonical

name

DistanceinAGraph

Date of creation

2013-03-22 12:31:32

Last modified on

2013-03-22 12:31:32

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 05C12

Synonym

distance

Related topic

Graph

Related topic

Path

Related topic

Diameter3

Related topic

PathConnected

Defines

diameter (of a graph)

Defines

radius (of a graph)

Defines

central vertex

Generated on Thu Feb 8 20:41:39 2018 by

LaTeXML
Hamming metric

Hamming metric

Let

x

=

(

x

1

,

x

2

,

x

3

,

…

,

x

n

)

,

y

=

(

y

1

,

y

2

,

y

3

,

…

,

y

n

)

be bit patterns, that is, vectors consisting of zeros and ones.

The

Hamming distance

d

H

⁢

(

u

,

v

)

defined as

∑

j

=

1

n

|

x

i

-

y

i

|

is equal to the number of positions where the bit patterns are differents.

For instance, if

u

=

(

0

,

1

,

1

,

0

,

1

,

0

,

1

)

and

v

=

(

1

,

0

,

1

,

0

,

1

,

0

,

1

)

then

d

H

⁢

(

u

,

v

)

=

|

0

-

1

|

+

|

1

-

0

|

+

|

1

-

1

|

+

|

0

-

0

|

+

|

1

-

1

|

+

|

0

-

0

|

+

|

1

-

1

|

=

3

because

u

and

v

have different bits at three positions.

The Hamming distance holds the

properties

of a metric (otherwise it would not be truly a

distance

):

•

d

H

⁢

(

x

,

y

)

≥

0

for any

x

,

y

.

•

d

H

⁢

(

x

,

y

)

=

0

if and only if

x

=

y

.

•

d

H

⁢

(

x

,

y

)

=

d

H

⁢

(

y

,

x

)

for any

x

,

y

.

•

d

H

⁢

(

x

,

y

)

≤

d

H

⁢

(

x

,

z

)

+

d

H

⁢

(

z

,

y

)

for any

x

,

y

,

z

.

If we realize that

d

H

is counting something (positions where bits differ), then it’s clear that

d

H

can never be

negative

. Also,

d

H

⁢

(

x

,

x

)

=

0

because a bit pattern has no different bits respect to itself, and if two bit patterns coincide on each position, they are indeed the same pattern, which proves the second property. The third condition also follows from the trivial fact that if

x

differs at some position from

y

, then

y

differs at the sae position from

x

.

We are left to prove the last condition (trangle inequality).
If

x

=

(

x

1

,

x

2

,

x

3

,

…

,

x

n

)

y

=

(

y

1

,

y

2

,

y

3

,

…

,

y

n

)

z

=

(

z

1

,

z

2

,

z

3

,

…

,

z

n

)

then

d

H

⁢

(

x

,

y

)

counts at how many places does

x

differ from

y

. For instance, suppose that

x

3

≠

y

3

. This means that the third bits are different, which adds

1

to the whole sum

d

H

⁢

(

x

,

y

)

.

Now, if

x

3

≠

y

3

it cannot happen that

x

3

=

z

3

and

z

3

=

y

3

at the same , so we have that

x

3

≠

z

3

or

z

3

≠

y

3

. In either case, the sum

d

H

⁢

(

x

,

z

)

+

d

H

⁢

(

z

,

y

)

also increases by one.

So, for each mismatch that increases

d

H

⁢

(

x

,

y

)

by one,

d

H

⁢

(

x

,

z

)

+

d

H

⁢

(

z

,

y

)

also increases by one. We conclude that

d

H

⁢

(

x

,

y

)

≤

d

H

⁢

(

x

,

z

)

+

d

H

⁢

(

z

,

y

)

.

Title

Hamming metric

Canonical

name

HammingMetric

Date of creation

2013-03-22 14:59:37

Last modified on

2013-03-22 14:59:37

Owner

drini (3)

Last modified by

drini (3)

Numerical id

7

Author

drini (3)

Entry type

Definition

Classification

msc 05C12

Classification

msc 94C99

Synonym

Hamming distance

Synonym

Hamming metric

Related topic

HammingDistance

Generated on Thu Feb 8 20:41:42 2018 by

LaTeXML
bipartite graph

bipartite graph

A

bipartite graph

is a graph with a

chromatic number

of

2

.

The following graph, for example, is bipartite:

\xymatrix

⁢

A

⁢

\ar

⁢

@

-

[

r

⁢

r

⁢

r

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

\ar

⁢

@

-

[

d

⁢

d

⁢

d

]

⁢

&

⁢

&

⁢

&

⁢

B

⁢

&

⁢

E

⁢

&

⁢

F

⁢

\ar

⁢

@

-

[

u

⁢

r

]

⁢

\ar

⁢

@

-

[

l

]

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

&

⁢

H

⁢

\ar

⁢

@

-

[

u

]

⁢

\ar

⁢

@

-

[

r

]

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

G

⁢

&

⁢

D

⁢

&

⁢

&

⁢

&

⁢

C

⁢

\ar

⁢

@

-

[

u

⁢

u

⁢

u

]

⁢

\ar

⁢

@

-

[

l

⁢

l

⁢

l

]

⁢

\ar

⁢

@

-

[

u

⁢

l

]

One way to think of a bipartite graph is by partitioning the vertices into two

disjoint sets

where vertices in one set are

adjacent

only to vertices in the other set. In the above graph, this may be more obvious with a different :

\xymatrix

⁢

A

⁢

\ar

⁢

@

-

[

r

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

⁢

d

]

⁢

&

⁢

E

⁢

F

⁢

\ar

⁢

@

-

[

r

⁢

u

]

⁢

\ar

⁢

@

-

[

r

]

⁢

\ar

⁢

@

-

[

r

⁢

d

⁢

d

]

⁢

&

⁢

B

⁢

H

⁢

\ar

⁢

@

-

[

r

⁢

u

⁢

u

]

⁢

\ar

⁢

@

-

[

r

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

D

⁢

C

⁢

\ar

⁢

@

-

[

r

⁢

u

⁢

u

]

⁢

\ar

⁢

@

-

[

r

⁢

u

]

⁢

\ar

⁢

@

-

[

r

]

⁢

&

⁢

G

The two subsets are the two columns of vertices, all of which have the same colour.

A graph is bipartite if and only if all its cycles have even length. This is easy to see intuitively: any path of odd length on a bipartite must end on a vertex of the opposite colour from the beginning vertex and hence cannot be a cycle.

Title

bipartite graph

Canonical name

BipartiteGraph

Date of creation

2013-03-22 12:17:07

Last modified on

2013-03-22 12:17:07

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 05C15

Synonym

bipartite

Generated on Thu Feb 8 20:41:44 2018 by

LaTeXML
chromatic number

chromatic number

The

chromatic number

of a graph is the minimum number of colours required to colour it.

Consider the following graph:

\xymatrix

⁢

A

⁢

\ar

⁢

@

-

[

r

]

⁢

&

⁢

B

⁢

\ar

⁢

@

-

[

r

]

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

C

⁢

\ar

⁢

@

-

[

r

]

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

F

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

D

⁢

\ar

⁢

@

-

[

r

]

⁢

&

⁢

E

⁢

&

This graph has been coloured using

3

colours. Furthermore, it’s clear that it cannot be coloured with fewer than

3

colours, as well: it contains a

subgraph

(

B

⁢

C

⁢

D

) that is

isomorphic

to the

complete graph

of

3

vertices. As a result, the chromatic number of this graph is indeed

3

.

This example was easy to solve by inspection. In general, however, finding the chromatic number of a large graph (and, similarly, an optimal

colouring

) is a very difficult (NP-hard) problem.

Title

chromatic number

Canonical

name

ChromaticNumber

Date of creation

2013-05-16 21:23:44

Last modified on

2013-05-16 21:23:44

Owner

mathcam (2727)

Last modified by

unlord (1)

Numerical id

9

Author

mathcam (1)

Entry type

Definition

Classification

msc 05C15

Related topic

ChromaticNumberAndGirth

Related topic

FourColorConjecture

Related topic

ChromaticPolynomial

Related topic

ChromaticNumberOfASpace

Generated on Thu Feb 8 20:41:46 2018 by

LaTeXML
chromatic number and girth

chromatic number and girth

A famous

theorem

of P. Erdős

1

1

See the very readable P. Erdős,

and

probability

, Canad J. Math. 11 (1959), 34–38.

.

Theorem 1

For any

natural numbers

k

and

g

, there exists a graph

G

with

chromatic number

χ

⁢

(

G

)

≥

k

and girth

girth

⁡

(

G

)

≥

g

.

Obviously, we can easily have graphs with high chromatic numbers. For

instance

, the

complete graph

K

n

trivially has

χ

⁢

(

K

n

)

=

n

; however

girth

⁡

(

K

n

)

=

3

(for

n

≥

3

). And the

cycle graph

C

n

has

girth

⁡

(

C

n

)

=

n

, but

χ

⁢

(

C

n

)

=

{

1

n

=

1

2

n

even

3

otherwise.

It seems intuitively plausible that a high chromatic number occurs because of short, “local” cycles in the graph; it is hard to envisage how a graph with no short cycles can still have high chromatic number.

Instead of envisaging, Erdős’ proof shows that, in some appropriately chosen

probability space

on graphs with

n

vertices, the probability of choosing a graph which does

not

have

χ

⁢

(

G

)

≥

k

and

girth

⁡

(

G

)

≥

g

tends to zero as

n

grows. In particular, the desired graphs exist.

This seminal paper is probably the most famous application of the

probabilistic method

, and is regarded by some as the

foundation

of the method.

2

2

However, as always, with the benefit of hindsight we can see that the probabilistic method had been used before, e.g. in various applications of Sard’s theorem. This does nothing to diminish from the importance of the clear statement of the tool.

Today the probabilistic method is a standard tool for

combinatorics

. More constructive methods are often preferred, but are almost always much harder.

Title

chromatic number and girth

Canonical name

ChromaticNumberAndGirth

Date of creation

2013-03-22 12:46:03

Last modified on

2013-03-22 12:46:03

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 05C15

Classification

msc 05C38

Classification

msc 05C80

Related topic

Girth

Related topic

ChromaticNumber

Generated on Thu Feb 8 20:41:48 2018 by

LaTeXML
colouring problem

colouring problem

The

colouring problem

is to assign a

colour

to every vertex of a graph such that no two

adjacent vertices

have the same colour. These colours, of course, are not necessarily colours in the optic sense.

Consider the following graph:

\xymatrix

⁢

A

⁢

\ar

⁢

@

-

[

r

]

⁢

&

⁢

B

⁢

\ar

⁢

@

-

[

r

]

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

C

⁢

\ar

⁢

@

-

[

r

]

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

F

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

D

⁢

\ar

⁢

@

-

[

r

]

⁢

&

⁢

E

⁢

&

One potential

colouring

of this graph is:

\xymatrix

⁢

A

⁢

\ar

⁢

@

-

[

r

]

⁢

&

⁢

B

⁢

\ar

⁢

@

-

[

r

]

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

C

⁢

\ar

⁢

@

-

[

r

]

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

F

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

&

⁢

D

⁢

\ar

⁢

@

-

[

r

]

⁢

&

⁢

E

⁢

&

A

and

C

have the same colour;

B

and

E

have a second colour; and

D

and

F

have another.

Graph colouring problems have many applications in such situations as scheduling and

matching

problems.

Title

colouring problem

Canonical name

ColouringProblem

Date of creation

2013-03-22 12:17:00

Last modified on

2013-03-22 12:17:00

Owner

vampyr (22)

Last modified by

vampyr (22)

Numerical id

7

Author

vampyr (22)

Entry type

Topic

Classification

msc 05C15

Synonym

coloring problem

Synonym

colour

Synonym

color

Synonym

graph colouring

Synonym

graph coloring

Generated on Thu Feb 8 20:42:22 2018 by

LaTeXML