An introduction to graph theory - Indiana State University

Bipartite Graphs

Definition
A graph G = (V , E ) is bipartite if
such that each edge of G has one
point in W .
Example
For which n, Cn is a bipartite ?

Arash Rafiey

f V can be partitioned into U, W
e end point in U and one end

An introduction to graph theory

Bipartite Graphs

Definition
A graph G = (V , E ) is bipartite if
such that each edge of G has one
point in W .
Example
For which n, Cn is a bipartite ?
For which value of n, Kn is bipart

Arash Rafiey

f V can be partitioned into U, W
e end point in U and one end

tite ?

An introduction to graph theory

Bipartite Graphs

Definition
A graph G = (V , E ) is bipartite if
such that each edge of G has one
point in W .
Example
For which n, Cn is a bipartite ?
For which value of n, Kn is bipart
For which value of n, Pn is biparti

Arash Rafiey

f V can be partitioned into U, W
e end point in U and one end

tite ?
ite ?

An introduction to graph theory

Bipartite Graphs

Definition
A graph G = (V , E ) is bipartite if
such that each edge of G has one
point in W .

Example
For which n, Cn is a bipartite ?
For which value of n, Kn is bipart
For which value of n, Pn is biparti
For which value of n, Qn is bipart

Arash Rafiey

f V can be partitioned into U, W
e end point in U and one end

tite ?
ite ?
tite ?

An introduction to graph theory

Definition
For positive integers n, m, the com
the following vertex and edge sets

V (Kn,m) = {u1, u2, . . . , un} ∪
E (Kn,m) = {ui vj |1 ≤ i ≤ n, 1

K
2,3

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mplete bipartite graph Kn,m has
s:
∪ {v1, v2, . . . , vm}
1 ≤ j ≤ m}

K
3,3

An introduction to graph theory

Degree Sequence

Give any graph, we can obtain the
of its vertices v1, v2, . . . , vn.

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e degree sequence (d1, d2, . . . , dn)

An introduction to graph theory

Degree Sequence

Give any graph, we can obtain the
of its vertices v1, v2, . . . , vn.
We are given a sequence (d1, d2, .
there exists a graph G whose degr
If this is the case then (d1, d2, . . .

Arash Rafiey

e degree sequence (d1, d2, . . . , dn)
. . . , dn). Can we decide whether
ree sequence is (d1, d2, . . . , dn) ?
, dn) is called a graphic sequence.

An introduction to graph theory

Degree Sequence

Give any graph, we can obtain the
of its vertices v1, v2, . . . , vn.
We are given a sequence (d1, d2, .
there exists a graph G whose degr
If this is the case then (d1, d2, . . .
(2, 2, 2) is graphic but (2, 3, 4, 3, 2

Arash Rafiey

e degree sequence (d1, d2, . . . , dn)
. . . , dn). Can we decide whether
ree sequence is (d1, d2, . . . , dn) ?
, dn) is called a graphic sequence.
2, 3) is not graphic.

An introduction to graph theory

Theorem

Suppose π = (d1, d2, . . . , dn) is a
n > d1 ≥ d2 ≥ · · · ≥ dn.

1 If π is graphic then there is a
V (G ) = {v1, v2, . . . , vn} and
of v1 are v2, v3, . . . , vd1+1.

2 π is graphic if and only if
(d2 − 1, d3 − 1, . . . , dd1+1 − 1

Proof.

Proof of (1). Since π is graphic, t
v1, v2, . . . , vn and degree sequence
We may assume that G = (V , E )
maximum where S = {v2, v3, . . . ,
neighborhood of v1). If v1 is adjac
we are done. Otherwise there exis
and hence there exits > d1 + 1 w

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sequence with

a graph G with
deg (vi ) = di and the neighbors

1, dd1+2, . . . , dn) is graphic.

there is a graph G with vertices
e π.
is such a graph that N(v1) ∩ S is
vd1+1} (N(v1) is the
cent to all the elements in S then
sts some vk such that v1vk ∈ E
where v1v ∈ E .

An introduction to graph theory

Proof.
Since k < , vk has as many neigh
neighbor vj that is not neighbor o
by removing edge v1v and adding
vk vj and adding edge vj v . G has
but N(v1) ∩ S increases in G , a c

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hbors as v and hence vk has a
of v . Now create new graph G
g edge v1vk and removing edge
s the same degree sequence as G
contradiction.

An introduction to graph theory

Proof.
Since k < , vk has as many neigh
neighbor vj that is not neighbor o
by removing edge v1v and adding
vk vj and adding edge vj v . G has
but N(v1) ∩ S increases in G , a c

Proof of (2). If π is graphic then
v1, v2, . . . , vn and degree sequence
v2, v3, . . . , vd1+1. If we remove v f
(d2 − 1, d3 − 1, . . . , dd1+1 − 1, dd1+
with graphic sequence (d2 − 1, d3
associated with graph G then we
it to the vertices of G with degre
This way we obtain G with degree

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hbors as v and hence vk has a
of v . Now create new graph G
g edge v1vk and removing edge
s the same degree sequence as G
contradiction.

by (1) there is G with vertices
e π where v1 is adjacent to
from G then we have a sequence
+2, . . . , dn). Conversely, if we start
− 1, . . . , dd1+1 − 1, dd1+2, . . . , dn)
e add a new vertex v and connect
ees d2 − 1, d3 − 1, . . . , dd1+1 − 1.
e sequence π.

An introduction to graph theory

Algorithm to detect graphic s

Graphic (n > d1 ≥ d2 ≥ d3 ≥ · · ·
1. while d1 > 0
2. Set (d1, d2, . . . , dn−1) be a n

(d2 − 1, d3 − 1, . . . , dd1+1 −
3. Set n = n − 1 and (d1, d2, . .
4. if di < 0 then output NO e
5. else if d1 = 0 then output Y

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sequence

· ≥ dn)
non-decreasing permutation of
1, dd1+2, . . . , dn)
. . , dn) = (d1, d2, . . . , dn)
exit
YES exit

An introduction to graph theory

Example
Initial Sequence : (4,4,3,3,2,2)

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An introduction to graph theory

Example
Initial Sequence : (4,4,3,3,2,2)
(1) : (3,2,2,1,2)

Arash Rafiey

An introduction to graph theory

Example
Initial Sequence : (4,4,3,3,2,2)
(1) : (3,2,2,1,2)
Sort : (3,2,2,2,1)

Arash Rafiey

An introduction to graph theory

Example
Initial Sequence : (4,4,3,3,2,2)
(1) : (3,2,2,1,2)
Sort : (3,2,2,2,1)
(2) : (1,1,1,1)

Arash Rafiey

An introduction to graph theory

Example
Initial Sequence : (4,4,3,3,2,2)
(1) : (3,2,2,1,2)
Sort : (3,2,2,2,1)
(2) : (1,1,1,1)
(3) : (0,1,1)

Arash Rafiey

An introduction to graph theory

Example
Initial Sequence : (4,4,3,3,2,2)
(1) : (3,2,2,1,2)
Sort : (3,2,2,2,1)
(2) : (1,1,1,1)
(3) : (0,1,1)
Sort (1,1,0)

Arash Rafiey

An introduction to graph theory

Example
Initial Sequence : (4,4,3,3,2,2)
(1) : (3,2,2,1,2)
Sort : (3,2,2,2,1)
(2) : (1,1,1,1)
(3) : (0,1,1)
Sort (1,1,0)
(4) : (0,0) True

Arash Rafiey

An introduction to graph theory

Example
Initial Sequence : (4,4,3,3,2,2)
(1) : (3,2,2,1,2)
Sort : (3,2,2,2,1)
(2) : (1,1,1,1)
(3) : (0,1,1)
Sort (1,1,0)
(4) : (0,0) True
Is the sequence (5, 5, 3, 5, 3, 3, 3, 3

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3) is graphic ?

An introduction to graph theory

Subgraphs

Definition
Let G = (V , E ) be a graph. Grap
G if V ⊆ V and E ⊆ E .

Arash Rafiey

ph H = (V , E ) is a subgraph of

An introduction to graph theory

Subgraphs

Definition
Let G = (V , E ) be a graph. Grap
G if V ⊆ V and E ⊆ E .
C5 is a subgraph of K6, and K1, K

Arash Rafiey

ph H = (V , E ) is a subgraph of
K2, . . . , K5 are all subgraph of K6.

An introduction to graph theory

Graph Isomorphism

Definition
Let G = (V , E ), G = (V , E ) tw
a one-to-one function.

1 f preserve adjacency if for

Arash Rafiey

wo graphs. Suppose f : V → V is
every uv ∈ E , f (u)f (v ) ∈ E .

An introduction to graph theory

Graph Isomorphism

Definition
Let G = (V , E ), G = (V , E ) tw
a one-to-one function.

1 f preserve adjacency if for
2 f preserve non-adjacency if

u, v then f (u), f (v ) are non-a

Arash Rafiey

wo graphs. Suppose f : V → V is
every uv ∈ E , f (u)f (v ) ∈ E .
f for every non adjacent vertices
adjacent.

An introduction to graph theory

Graph Isomorphism

Definition
Let G = (V , E ), G = (V , E ) tw
a one-to-one function.

1 f preserve adjacency if for
2 f preserve non-adjacency if

u, v then f (u), f (v ) are non-a
3 f is a graph isomorphism f

preserve both adjacency and
write G ∼= G .

Arash Rafiey

wo graphs. Suppose f : V → V is
every uv ∈ E , f (u)f (v ) ∈ E .
f for every non adjacent vertices
adjacent.
from G to G if it is bijective and
non-adjacency. In this case we

An introduction to graph theory