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Graph theory

Chapter 1
: Introduction

Definitions:

Order

Number of vertices

Size

Number of edges

Induced subgraph

A subgraph F of a graph G whenever u and v are vertices of F and uv
is an
edge in G, then it is an edge in F.

Trail

Walk with no edge traverse
d more than once.

Path

Walk with no vertice traversed more than once.

Circuit

Closed trail with length 3 or more.

Geodesic

A path from u to v with length d(u,v).

Diameter

The greatest distance between any two vertices in a graph.

Complete

A graph is comple
te if every two distinct vertices of G are adjacent.

Complement of graph

The complement has every edge graph G does not have

Bipartite graph

A graph of 2 components, each component is a partite set

Complete bipartite

If every vertice of a component is conn
ected to the vertices of the other
component

k
-
partite graph

A graph of k components

Join

The union between a graph G en H where all vertices are connected to the
vertices of the other graph

Cartesian product

2 graphs multiplied


Theorems:

Theorem 1.6:

I
f a graph G contains a u
-
v walk of length l, then

G contains a u
-
v path of length
at most l

Theorem 1.7:

Let R be the relation defined on the vertex set of a praph G by u R v, where u,v є
V(G) such that u R v and u R w. Hence G contains a u
-
v path P’ and a

v
-
w path P’’.
As we have seen earlier, following P’ by P’’ produces a u
-
w walk W. By Theorem
1.6, G contains a u
-
w path and so u R w.

Theorem 1.8:

Let G be a graph of order 3 or more. If G contains two distinct vertices u and v
such G
-
u and G
-
v are connec
ted, then G itself is connected

Theorem 1.9:

If G is a connected graph of order 3 or more, then G contains two distinct
vertices u and v such that G
-
u and G
-
v are connected.

Theorem 1.10:

Let G be a graph of order 3 or more. Then G is connected if and only

if G
contains two distinct vertices u and v such that G
-
u and G
-
v are connected

Theorem 1.11:

If G is a disconnected graph, then the complement of G is connected

Theorem 1.12:

A non
-
trivial graph G is a bipartite graph if and only if G contains no odd cyc
les


Chapter 2: Degrees


Definitions:

Degree of a vertex

Number of edges incident with an edge

Neighbors


Two adjacent vertices

Neighborhood


All neighbors

Even (odd) vertex

Vertex with even(odd) degree

Sharpness

The sharpness of a theory means that some

statement could is not the bound of the statement,
for example bla ≤ n
-
1 is true, but is bla≤ n
-
2 also true.

r
-
regular graph

A graph where every vertex has the degree r

Petersen graph

A 3
-
regular graph of degree 10(see picture)


Harary graphs

r
-
regular gr
aphs H
r,n

with order n and the properties of theorem 2.6

Degree sequence

All degrees of the vertices of a graph sorted in a sequence.

Graphical

A degree sequence is graphical if a graph can be constructed from it

Adjacency matrix

Matrix with n columns and
n rows with where

A
ij

=

Incidence matrix

n x m matrix where




B
ij

=

Equal walks

Two walks are equal if they are equal term to term


Theorems:

Theorem 2.1


If G is a graph of size m, then


Corollary 2.3


Every graph has a
n even number of odd vertices

Theorem 2.4

Let G be a graph of order n. If







for every two non
-
adjacent vertices u and v of G, then G is connected and
diam(G) ≤ 2.

Corollary 2.5


If G is a graph of order n with δ(G) ≥ (n
-
1)/2, then G is c
onnected

Theorem 2.6

Let r and n be integers with 0 ≤ r ≤ n
-
1. There exists an r
-
regular graph of
order n if and only if at least one of r and n are even.

Theorem 2.7

For every graph G and every integer r ≥ Δ(G), there exists an r
-
regular graph H
containin
g G as an induced subgraph.

Theorem 2.10

A non
-
increasing sequence s : d
1
, d
2
,…., d
n

(n≥2) of non
-
negative integers,
where d
1
≥1, is graphical if and only if the sequence




s
1

: d
2



1, d
3



1, d
d1+1
-
1, d
d1 + 2
,…d
n



is graphical.

Theorem 2.13

Let G be a

graph with vertex set V(G) = {v
1
, v
2
, … , v
n
} and adjacency matrix A
= [
a
ij
]
. Then the entry a
ij
(k)
in row i and column j of A
k

is the number of distinct
v
i



v
j

walks of length k in G.


Chapter 4: Trees

Definitions:

Bridge



An edge uv is a brige when G


uv is disconnected, while G is connected


Tree



A tree is an acyclic graph

Caterpillar


A tree of order 3 or more

Spine



A caterpillar without end
-
vertices

Forest



A cyclic graph where each components is a tree

Isomorphic Graphs

G
raphs t
hat are struct
urally equivalent

Spanning subgraph

Subgraph H is a spanning subgraph of G if it contains every vertex of G.

Spanning tree


If H is a tree it is called a spanning tree

Cost/weight


Edge can have certain costs or weights

Weight of a graph

The weight of a g
raph is all weights summed up

Minimum spanning tree: Spanning tree with minimum weight

Algorithms for mst
-
problem:

-

Kruskal’s algorithm:

For a connected weighted graph G, a spanning tree T of G constructed follows: For the first
edge e
1

of T, we select any
edge of G of minimum weight and for the second edge e
2

of T, we
select any remaining edge of minimum weight. For the third edge e
3

of T, we choose any
remaining edge of G of minimum weight that does not produce a cycle with the previously
selected edges. W
e continue in this manner until a spanning tree is produced.

-

Prim’s algorithm:

For a connected weighted graph G, a spanning tree T of is constructed as follows: For an
arbitrary vertex u for G, an edge minimum weight incident with u is selected as first ed
ge e
1

of T. For subsequent edges e
2
, e
3
,..,e
n
-
1

, we select an edge of minimum weight among those
edges having exactly one of its vertices incident with an edge already selected.


Theorems:

Theorem 4.1


An edge e of a graph G is a bridge if and only if e l
ies on no cycle of G

Theorem 4.2

A graph G is a tree if and only if every two vertices of G are connected by a
unique path

Theorem 4.3

Every nontrivial tree has at least two end
-
vertices

Theorem 4.4

Every tree of order n has size n
-
1

Corollary 4.6

Every fo
rest of order n with k components has size n


k

Theorem 4.7

The size of every connected graph of order n is at least n
-
1

Theorem 4.8

Let G be a graph of order n and size m. If G satisfies any two properties:

1.

G is connected,

2.

G is acyclic,

3.

M = n
-
1,

then G i
s a tree.

Theorem 4.9

Let T be a tree of order k. If G is a graph with δ(G) ≥ k
-
1, then T is isomorphic
to some subgraph of G

Theorem 4.10

Every connected graph contains a spanning tree

Theorem 4.11

Kruskal’s algorithm produces a minimum spanning tree in a connecte
d
weighted graph.


Chapter 6 Traversability


Definitions:

Eulerian circuit

A circuit C in a graph G which contains every edge exactly once

Eulerian graph

A graph with an eulerian circuit

Eulerian trail

A trail that contains all edges in a graph

Hamiltonian

cycles

A cycle that contains all vertices in a graph

Hamiltonian graph

A graph that contains a Hamiltonian cycle

Hamiltonian path

A path that contains all vertices in a graph

Closure; C(g)

A graph G with order n, the closure of G is the graph obtained by
recursively
joining pairs of non
-
adjacent, whose degree sum is at least n, until no such
pair remains.

Cut
-
vertex

A vertex in a connected graph G if G


v is disconnected


Theorems:

Theorem 6.1

A nontrivial connected graph G is Eulerian if and only if ever
y vertex has an
even degree.

Corollary 6.2

A connected graph G contains an Eulerian trail if and only if exactly two
vertices of G have odd degrees. Furthermore, each Eulerian trail of G begins
or ends at one of these odd vertices and ends at the other

The
orem 6.3

Let G and H be nontrivial connected graphs. Then G x H is Eulerian if and only
if both G and H are Eulerian or every vertex of G and H is odd.

Theorem 6.4

The Petersen graph is not Hamiltonian


Theorem 6.5

If G is a Hamiltonian graph, then for ev
ery nonempty proper set S of vertices
of G,




k
(G



S
) ≤ |S|
.

(k(G) is the number of components in G)

Theorem 6.6

Let G be a graph of order n ≥ 3.
If




Deg u + deg v ≥ n


for each pair u, v of nonadjacent vertices of G, then G is Hamiltonian.


Corollary 6.7

Let G be a graph of order n ≥ 3. If deg v ≥ n/2 f
or each vertex v of G, the G is
Hamiltonian.

Theorem 6.8

Let u and v be nonadjacent vertices in a graph G of order n such that deg u +
deg v ≥ n. Then G + uv is Hamiltonian if and only if G is Hamiltonian.

Theorem 6.9

A graph is Hamiltonian if and only if

its closure is Hamiltonian

Corollary 6.10

If G is a graph of order at least 3 such that C(G) is complete, then G is
Hamiltonian

Theorem 6.11

Let G be a graph of order n ≥ 3. If for every integer j with 1 ≤ j ≤

, the
number of vertices of G w
ith degree at most j is less than j, then G is
Hamiltonian.


Chapter 8

Matchings and factorization


Definitions:

Independent

A set of edges
(vertices)

is independent if no two edges
(vertices)

are adjacent

Matching

Let G be a bipartite graph with partite set
s U and W, where r = |U| ≤ |W|. A
matching in G is therefore a set M = {e
1
,e
2
,…,e
k
} of edges, where e
i

= u
i
w
i

for 1
≤ i ≤ k such that u
1
, u
2
, … u
k

are k distinct vertices of U and w
1
, w
2
, … , w
k

are k
distinct vertices of W.

Neighborly

Given neighborhood
N(X) a set U is neighborly if |N(X)|≥ |X| where for every
nonempty subset X of U

and N(X) the union of all neighbors x є X
.
If the
neighborhood of a subset of X of U is bigger or equal to the subset X for
every

subset of X of U, then U is neighborly.

Cove
r

A vertex and an incident edge cover each other

Edge cover

A graph without isolated vertices is a set of edges of G that covers all vertices
of G


α(G)

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β

(G)

Vertex independence
number

Maximum number of vertices, no
two which are adjacent

α
1
(G)

Edge covering number

Minimum number of edges that
cover all vertices of G

β

1
(G)

Edge independence number

Maximum number of edges, no
two which are adjacent




Theorems:


Theorem 8.3

Let G be a bipartite graph with partite sets U and W such that r = |U| ≤ |W|.
Then G contains a matching of cardinality r if and only if U is neighborly.

Theorem 8.4

A collection {S
1
, S
2
, … , S
n
} of a nonempty finite sets has a system of distin
ct
representatives if and only if for each integer k with 1 ≤ k ≤ n, the union of
any k of these sets contains at least k elements.

Theorem 8.5

In a collection of r women and s men, where 1 ≤ r ≤ s, a total of r marriages

(Marriage Theorem)


between acquai
nted couples is possible if and only if for each integer k with
1 ≤ k ≤ r, every subset of k women is collectively acquainted with at least k
men.

Theorem 8.6

Every r
-
regular bipartite graph (r ≥ 1) has a perfect matching

Theorem 8.7

For every graph G of

order n containing no isolated vertices,





α
1
(G) +
β

1
(G) = n
.

Theorem 8.8

For every graph G of order n containing no isolated vertices,





α
(G) +
β

(G) = n.

8.14

A graph G without isolated vertices contains a has a perfect a perfect
matching if and o
nly if
α
1
(G) =
β
1
(G)
.


For 8.2 see handout Hungarian method


Chapter 9 Planarity


Definitions:

Planar graph

A graph that can we drown without two edges crossing each other

Plane graph

A graph is drawn in a plane without no two edges of G cross

Regions

The
connected pieces of a plane graph

Exterior region

The unbounded part of every plane graph

Boundary of a region

The subgraph of all vertices and edges that are
incident with the region

Maximal planar

G is planar, but the addition of an edge between
two non
adjacent vertices of G makes it nonplanar

Subdivision

One or more vertices of degree 2 are inserted
into one or more edges of G(see picture)


Theorems:

Theorem 9.1

If G is a connected plane graph of order n, size m, and having r regions, then
n


m + r = 2

Theorem 9.2

If G is a planar graph of order n ≥ 3 and size m, then





m ≤ 3n


6.

Corollary 9.3

Every planar graph contains a vertex of
degree 5 or less

Corollary 9.4

The complete graph K
5

is nonplanar
(see
picture)

Theorem 9.5

The graph K
3,3

is nonpla
nar

(see picture)

Theorem 9.7

A graph G is planar if and only if G does not contain K
5
, K
3,3

or a subdivision of
K
5

or K
3,3

as a subgraph
.



Chapter 10 Coloring


Definitions:

Dual

If
you represent a map by a graph it is called a dual

Coloring

Assigning col
ors to regions, such that every adjacent vertex has a separate
color

Chromatic number

Smallest number of colors needed for coloring denoted as χ(G)

k
-
coloring

If it is possible to color G from a set of k colors, then G is said to be k
-
colorable

k
-
coloring

A coloring that uses k colors

Color classes

if G is k
-
chromatic then it is possible to divide G in k independent sets, these
are called color classes

Clique

A complete subgraph of graph G.

Clique number

The order of the largest clique, denoted as ω(G)

Sha
dow graph

Obtained from graph G by adding, for each vertex v of G, a new vertex v’,
called the
shadow vertex

of v, and joining v’ to the neighbors of v in G
.



Theorems:

Theorem 10.1

The chromatic number of every planar graph is at most 4(The Four Color
Th
eorem)

Theorem 10.2

A graph G has chromatic number 2 if and only if G is a nonempty bipartite
graph

Theorem 10.5


For every graph G of order n:

χ(G) ≥ ω(G) and χ(G) ≥

Theorem 10.7


For every graph:

χ(G) ≤ 1 + Δ(G)

Theorem 10.8


For every co
nnected graph G that is not an odd cycle or a complete graph







χ(G) ≤ Δ(G)

Theorem 10.9


For every graph G,

χ(G) ≤ 1 + max{δ(H)},





where the maximum is taken over all induced subgraphs H of G.

Theorem 10.10

For every integer k ≥ 3, there exists a tr
iangle
-
free graph with chromatic
number k.