graph theory

graph theory

Graph theory

is the branch of mathematics that concerns itself with graphs.

The

concept

of graph is extraordinarily simple, which explains the wide applicability of graph theory. It is usually agreed upon that graph theory proper was born in 1736, when Euler formalized the now-famous “

http://planetmath.org/node/2810

bridges of Königsberg” problem. Graph theory has now grown to touch almost every mathematical discipline, in one form or another, and it likewise borrows from elsewhere tools for its own problems. Anyone who delves into the topic will quickly see that the lifeblood of graph theory is the abundancy of tricky questions and clever answers. There are, of course, general results that systematize the subject, but we also find an emphasis on the solutions of substantial problems over building machinery for its own sake.

Graph theory representation of Königsberg bridges problem

For quite a long time graph theory was regarded as a branch of

topology

concerned with

1

-dimensional simplices, however this view has faded away. The only remainder of the topological past is the topological graph theory, a branch of graph theory that primarily deals with drawing of graphs on surfaces. The most famous achievement of topological graph theory is the proof of the

four-color conjecture

(every political map on the surface of a plane or a sphere can be colored into four colors given that each country consists of only one piece).

Now, a (finite) graph is usually thought of as a subset of pairs of elements of a

finite set

(called vertices), or more generally as a family of arbitrary sets in the case of

hypergraphs

. For

instance

,

Ramsey theory

as applied to graph theory deals with determining how disordered can graphs be. The

central

result here is the Ramsey’s

theorem

(

http://planetmath.org/RamseysTheorem2

) which states that one can always find many vertices that are either all

connected

or disconnected from each other, given that the graph is sufficiently large. The other result is Szemerédi regularity lemma.

The four-color conjecture mentioned above is one of the problems in

graph coloring

. There are many ways one can color a graph, but the most common are

vertex coloring

and

edge coloring

. In these type of

colorings

, one colors vertices (edges) of a graph so that no two vertices of the same color are

adjacent

(resp. no edges of the same color share a common vertex). The most common problem is to find a coloring using the fewest number of colors possible. Such problems often arise in scheduling problems.

Graph theory benefits greatly from interaction with other fields of mathematics. For example,

probabilistic methods

have become the standard tool in the arsenal of graph theorists, and

random graph

theory has grown into a full-fledged branch of its own. There are close, mathematical links between the

foundation

of

category theory

and
graph theory, as a

category

can be defined as a special type of meta-graph, and a

supercategory

may also have a meta-graph representation as a

meta-category

(

[

1

]

). Moreover, a graph can be regarded
as a one-dimensional

C

⁢

W

complex. Free groupoids can be employed to generate graphs using highly efficient computer programming; thus, the

Cayley graph

allows one to form a graph from a group, and the

transformation

groupoid

is a
way to form a groupoid from a group. On the other hand, the underlying graph of a transformation groupoid
is a way to form a graph from a groupoid. Moreover, an

isomorphism

graph theorem

proves that the underlying
graph of a transformation groupoid is isomorphic to the Cayley graph of the group whose action on itself forms the
transformation groupoid. Graphs also have several applications in theoretical physics, as for example in the case of Feynman diagrams (that are graphs). Furthermore, there are several

extensions

of the graph concept such as the hypergraph; thus, many of the definitions of types of graphs carry over to hypergraphs, even though a hypergraph is a

simple incidence structure

. Examples are presented under the hypergraph entry of

k

-uniform and

k

-regular hypergraphs, and the

connection

is made to matrices via the incidence matrix associated with a hypergraph.

There is an alternative approach to defining

abstract

graphs in terms of graph morphisms
in the category of graphs.

Definition 1.1

A

graph morphism

f

from graph

A

to graph

B

can be defined as a pair of

functions

f

E

:

A

E

→

B

E

, and

f

V

:

A

V

→

B

V

, preserving the source and target maps

s

,

t

:

A

E

→

A

V

, as well as the dual map

ϵ

:

A

V

→

A

E

, where

A

E

is the set of

graph edges

in graph

A

, and

A

V

is the set of

graph vertices

in graph

A

;

B

E

, and

B

V

are, respectively, the set of edges and the set of vertices of graph

B

.

References

1

S. Mac Lane and I. Moerdijk.

Sheaves in Geometry and Logic.

, Springer-Verlag:
New York, Berlin and Heidelberg (2000).

Title

graph theory

Canonical name

GraphTheory

Date of creation

2013-05-16 20:59:21

Last modified on

2013-05-16 20:59:21

Owner

karteef (1114)

Last modified by

unlord (1)

Numerical id

50

Author

karteef (1)

Entry type

Topic

Classification

msc 81Q30

Classification

msc 18G30

Classification

msc 05C50

Classification

msc 55U10

Classification

msc 18-00

Classification

msc 05C25

Classification

msc 05C99

Synonym

category theory

Related topic

Graph

Related topic

Hypergraph

Related topic

Category

Related topic

CategoryTheory

Related topic

EulersPolyhedronTheorem

Related topic

Digraph

Related topic

Tree

Related topic

BinaryTree

Related topic

ProbabilisticMethod

Related topic

HamiltonianCycle

Related topic

Tournament

Related topic

RamseysTheorem

Related topic

Coloring

Related topic

GraphTopology

Related topic

DirectedGraph

Related topic

DepthFirstSearch

Related topic

DijkstrasAlgorithm

Defines

graph morphism

Defines

abstract graph

Defines

category of graphs

Generated on Sat Feb 10 13:05:33 2018 by

LaTeXML