Graph theory

Graph Theory
Ch. 1. Fundamental Concept
1
Chapter 1
Fundamental Concept
1.1 What Is a Graph?
1.2 Paths, Cycles, and Trails
1.3 Vertex Degree and Counting
1.4 Directed Graphs

Graph Theory
2
The KÖnigsberg Bridge Problem
 Königsber is a city on the Pregel river in Prussia
 The city occupied two islands plus areas on both
banks
 Problem:
Whether they could leave home, cross every
bridge exactly once, and return home.
X
Y
Z
W

Graph Theory
3
A Model
 A vertex : a region
 An edge : a path(bridge) between two
regions
e1
e2
e3
e4
e6
e5
e7
Z
Y
X
W
X
Y
Z
W

Graph Theory
4
General Model
 A vertex : an object
 An edge : a relation between two objects
common
member
Committee 1 Committee 2

Graph Theory
5
What Is a Graph?
 A graph G is a triple consisting of:
– A vertex set V(G )
– An edge set E(G )
– A relation between an edge and a pair of
vertices
e1
e2
e3
e4
e6
e5
e7
Z
Y
X
W

Graph Theory
6
Loop, Multiple edges
 Loop : An edge whose endpoints are equal
 Multiple edges : Edges have the same pair of
endpoints
loop
Multiple
edges

Graph Theory
7
Simple Graph
 Simple graph : A graph has no loops or multiple
edges
loop
Multiple
edges
It is not simple. It is a simple graph.

Graph Theory
8
Adjacent, neighbors
 Two vertices are adjacent and are neighbors
if they are the endpoints of an edge
 Example:
– A and B are adjacent
– A and D are not adjacent
A B
C D

Graph Theory
9
Finite Graph, Null Graph
 Finite graph : an graph whose vertex set and
edge set are finite
 Null graph : the graph whose vertex set and
edges are empty

Graph Theory
10
Complement
 Complement of G: The complement G’ of a
simple graph G :
– A simple graph
– V(G’) = V(G)
– E(G’) = { uv | uv ∉E(G) }
G
u
v
w
x
y
G’
u
v
wx
y

Graph Theory
11
Clique and Independent set
 A Clique in a graph: a set of pairwise
adjacent vertices (a complete subgraph)
 An independent set in a graph: a set of
pairwise nonadjacent vertices
 Example:
– {x, y, u} is a clique in G
– {u, w} is an independent set G
u
v
wx
y

Graph Theory
12
Bipartite Graphs
 A graph G is bipartite if V(G) is the union of
two disjoint independent sets called partite
sets of G
 Also: The vertices can be partitioned into two
sets such that each set is independent
 Matching Problem
 Job Assignment Problem
Workers
Jobs
Boys
Girls

Graph Theory
13
Chromatic Number
 The chromatic number of a graph G,
written x(G), is the minimum number of colors
needed to label the vertices so that adjacent
vertices receive different colors
Red
Green
Blue
Blue
x(G) = 3

Graph Theory
14
Maps and coloring
 A map is a partition of the plane into
connected regions
 Can we color the regions of every map using
at most four colors so that neighboring
regions have different colors?
 Map Coloring → graph coloring
– A region → A vertex
– Adjacency → An edge

Graph Theory
15
Scheduling and graph Coloring 1
 Two committees can not hold meetings at the
same time if two committees have common
member
common
member

Graph Theory
16
 Model:
– One committee being represented by a
vertex
– An edge between two vertices if two
corresponding committees have common
member
– Two adjacent vertices can not receive the
same color
common
member

Graph Theory
17
 Scheduling problem is equivalent to graph
coloring problem
Common
MemberCommittee 1
Committee 2
Committee 3
Common Member
Different Color
No Common Member
Same Color OK
Same time slot OK

Graph Theory
18
Path and Cycle
 Path : a sequence of distinct vertices such
that two consecutive vertices are adjacent
– Example: (a, d, c, b, e) is a path
– (a, b, e, d, c, b, e, d) is not a path; it is a walk
 Cycle : a closed Path
– Example: (a, d, c, b, e, a) is a cycle
a b
c
de

Graph Theory
19
Subgraphs
 A subgraph of a graph G is a graph H such
that:
– V(H) ⊆ V(G) and E(H) ⊆ E(G) and
– The assignment of endpoints to edges in H is
the same as in G.

Graph Theory
20
Subgraphs
 Example: H1, H2, and H3 are subgraphs of G
c
d
a b
de
a b
c
de
H1
G
H3
H2
a b
c
de

Graph Theory
21
Connected and Disconnected
 Connected : There exists at least one path
between two vertices
 Disconnected : Otherwise
 Example:
– H1 and H2 are connected
– H3 is disconnected
c
d
a b
de
a b
c
d
eH1
H3H2

Graph Theory
22
Adjacency, Incidence, and Degree
 Assume ei is an edge whose endpoints are
(vj,vk)
 The vertices vj and vk are said to be adjacent
 The edge ei is said to be incident upon vj
 Degree of a vertex vkis the number of edges
incident upon vk. It is denoted as d(vk)
ei
vj
vk

Graph Theory
23
Adjacency matrix
 Let G = (V, E), |V| = n and |E|=m
 The adjacency matrix of G written A(G), is
the n-by-n matrix in which entry ai,j is the
number of edges in G with endpoints {vi, vj}.
a
b
c
d
e
w
x
y z
w x y z
0 1 1 0
1 0 2 0
1 2 0 1
0 0 1 0
w
x
y
z

Graph Theory
24
Incidence Matrix
 Let G = (V, E), |V| = n and |E|=m
 The incidence matrix M(G) is the n-by-m
matrix in which entry mi,j is 1 if vi is an endpoint
of ei and otherwise is 0.
a
b
c
d
e
w
x
y
z
a b c d e
1 1 0 0 0
1 0 1 1 0
0 1 1 1 1
0 0 0 0 1
w
x
y
z

Graph Theory
25
Isomorphism
 An isomorphism from a simple graph G to a
simple graph H is a bijection f:V(G)→V(H) such
that uv ∈E(G) if and only if f(u)f(v) ∈ E(H)
– We say “G is isomorphic to H”, written G ≅
H
HG
w
x z
y c d
ba
f1: w x y z
c b d a
f2: w x y z
a d b c

Graph Theory
26
Complete Graph
 Complete Graph : a simple graph whose
vertices are pairwise adjacent
Complete Graph

Graph Theory
27
Complete Bipartite Graph or Biclique
 Complete bipartite graph (biclique) is a
simple bipartite graph such that two vertices are
adjacent if and only if they are in different partite
sets.
Complete Bipartite Graph

Graph Theory
28
Petersen Graph 1.1.36
 The petersen graph is the simple graph
whose vertices are the 2-element subsets of a
5-element set and whose edges are pairs of
disjoint 2-element subsets

Graph Theory
29
 Assume: the set of 5-element be (1, 2, 3, 4, 5)
– Then, 2-element subsets:
(1,2) (1,3) (1,4) (1,5) (2,3) (2,4) (2,5) (3,4) (3,5)
(4,5)
Disjoint, so
connected
45: (4, 5)
12
34
15
23
45
35
13
14
24
25

Graph Theory
30
 Three drawings

Graph Theory
31
Theorem: If two vertices are non-adjacent in the
Petersen Graph, then they have exactly one common
neighbor. 1.1.38
Proof:
x, zx, y
No connection,
Joint, One common element.
u, v Since 5 elements totally,
5-3 elements left.
Hence, exactly one of this
kind.
3 elements in these vertices
totally

Graph Theory
32
Girth 1.1.39, 1.1.40
 Girth : the length of its shortest cycle.
– If no cycles, girth is infinite

Graph Theory
33
Girth and Petersen graph 1.1.39, 1.1.40
 Theorem: The Petersen Graph has girth 5.
Proof:
– Simple → no loop → no 1-cycle (cycle of length
1)
– Simple → no multiple → no 2-cycle
– 5 elements →no three pair-disjoint 2-sets → no
3-cycle
– By previous theorem, two nonadjacent vertices
has exactly one common neighbor → no 4-cycle
– 12-34-51-23-45-12 is a 5-cycle.

Graph Theory
34
Walks, Trails1.2.2
 A walk : a list of vertices and edges v0, e1, v1,
…., ek, vk such that, for 1 ≤ i ≤ k, the edge ei
has endpoints vi-1 and vi.
 A trail : a walk with no repeated edge.

Graph Theory
35
Paths 1.2.2
 A u,v-walk or u,v-trail has first vertex u and last
vertex v; these are its endpoints.
 A u,v-path: a u,v-trail with no repeated vertex.
 The length of a walk, trail, path, or cycle is its
number of edges.
 A walk or trail is closed if its endpoints are
the same.

Graph Theory
36
Lemma: Every u,v-walk contains a u,v-path 1.2.5
Proof:
 Use induction on the length of a u, v-walk W.
 Basis step: l = 0.
– Having no edge, W consists of a single vertex
(u=v).
– This vertex is a u,v-path of length 0.
to be continued

Graph Theory
37
Proof: Continue
 Induction step : l ≥ 1.
– Suppose that the claim holds for walks of
length less than l.
– If W has no repeated vertex, then its vertices
and edges form a u,v-path.

Graph Theory
38
Proof: Continue
 Induction step : l ≥ 1. Continue
– If W has a repeated vertex w, then deleting the
edges and vertices between appearances of w
(leaving one copy of w) yields a shorter u,v-
walk W’ contained in W.
– By the induction hypothesis, W ’ contains a
u,v-path P,and this path P is contained in W.
u
v
W
P
Delete

Graph Theory
39
Components 1.2.8
 The components of a graph G are its
maximal connected subgraphs
 A component (or graph) is trivial if it has no
edges; otherwise it is nontrivial
 An isolated vertex is a vertex of degree 0
r
q
s u v w
t p x
y z

Graph Theory
40
Proof:
– An n-vertex graph with no edges has n
components
– Each edge added reduces this by at most 1
– If k edges are added, then the number of
components is at least n - k
Theorem: Every graph with n vertices and k
edges has at least n-k components 1.2.11

Graph Theory
41
Theorem: Every graph with n vertices and k edges
has at least n-k components 1.2.11
 Examples:
n =2, k =1,
1 component
n =3, k =2,
1 component
n =6, k =3,
3 components
n =6, k =3,
4 components

Graph Theory
42
Cut-edge, Cut-vertex 1.2.12
 A cut-edge or cut-vertex of a graph is an
edge or vertex whose deletion increases the
number of components
Not a Cut-vertexCut-edge
Cut-edge
Cut-vertex

Graph Theory
43
Cut-edge, Cut-vertex 1.2.12
 G-e or G-M : The subgraph obtained by
deleting an edge e or set of edges M
 G-v or G-S : The subgraph obtained by
deleting a vertex v or set of vertices S
e
G-eG

Graph Theory
44
Induced subgraph 1.2.12
 An induced subgraph :
– A subgraph obtained by deleting a set of
vertices
– We write G[T] for G-T’, where T’ =V(G)-T
– G[T] is the subgraph of G induced by T
 Example:
– Assume T:{A, B, C, D}
BA
C D
E
BA
C D
G[T]
G

Graph Theory
45
 More Examples:
– G2 is the subgraph of G1 induced by
(A, B, C, D)
– G3 is the subgraph of G1 induced by (B, C)
– G4 is not the subgraph induced by (A, B, C, D)
BA
C D
E
BA
C D
B
C
BA
C D
G1 G2
G3 G4

Graph Theory
46
 A set S of vertices is an independent set if and
only if the subgraph induced by it has no edges.
– G3 is an example.
BA
C D
E
B
C
G1
G3

Graph Theory
47
Theorem: An edge e is a cut-edge if and only if e
belongs to no cycles. 1.2.14
Proof :1/2
 Let e= (x, y) be an edge in a graph G and H be the
component containing e.
– Since deletion of e effects no other component,
it suffices to prove that H-e is connected if and
only if e belongs to a cycle.
 First suppose that H-e is connected.
– This implies that H-e contains an x, y-path,
– This path completes a cycle with e.

Graph Theory
48
Theorem: An edge e is a cut-edge if and only if e
belongs to no cycles. 1.2.14
Proof :2/2
 Now suppose that e lies in a cycle C.
– Choose u, v∈V(H)
• Since H is connected, H has a u, v-path P
– If P does not contain e
• Then P exists in H-e
– Otherwise (P contains e)
• Suppose by symmetry that x is between u and y
on P
• Since H-e contains a u, x-path along P,
the transitivity of the connection relation implies
that H-e has a u, v-path
– We did this for all u, v ∈ V(H), so H-e is
connected.

Graph Theory
49
Theorem: An edge e is a cut-edge if and only
if e belongs to no cycles. 1.2.14
 An Example:
u
x y
e
P
C
v

Graph Theory
50
Lemma: Every closed odd walk contains an odd cycle
Proof:1/3
 Use induction on the length l of a closed odd walk W.
 l=1. A closed walk of length 1 traverses a cycle of
length 1.
 We need to prove the claim holds if it holds for
closed odd walks shorter than W.

Graph Theory
51
Proof: 2/3
 Suppose that the claim holds for closed odd walks shorter
than W.
 If W has no repeated vertex (other than first = last),
then W itself forms a cycle of odd length.
 Otherwise, (W has repeated vertex )
– Need to prove: If repeated, W includes a
shorter closed odd walk. By induction, the
theorem hold

Graph Theory
52
Proof: 3/3
– If W has a repeated vertex v, then we view W
as starting at v and break W into two v,v-walks
• Since W has odd length, one of these is odd and
the other is even. (see the next page)
• The odd one is shorter than W, by induction
hypothesis, it contains an odd cycle, and this
cycle appears in order in W
Even
v
Odd
Odd = Odd + Even

Graph Theory
53
Theorem: A graph is bipartite if and only if it has
no odd cycle. 1.2.18
 Examples:
BA
D C
A
C
B
D
A
C
B
D
F
E
A
C
B
D
E F

Graph Theory
54
Theorem: A graph is bipartite if it has no odd cycle. 1.2.18
Proof: (sufficiency1/3)
 Let G be a graph with no odd cycle.
 We prove that G is bipartite by constructing a bipartition
of each nontrivial component H.
 For each v ∈ V (H ), let f (v ) be the minimum length of a
u, v -path. Since H is connected, f (v ) is defined for
each v ∈ V (H ) .
u
v
'v
( ')f v
( )f v

Graph Theory
55
 Let X={v ∈ V (H ): f (v ) is even} and Y={v ∈ V(H ): f (v )
is odd}
 An edge v, v’ within X (or Y ) would create a closed odd
walk using a shortest u, v-path, the edge v, v’ within X (or
Y ) and the reverse of a shortest u, v’-path.
u
v
'v
A closed odd walk using
1) a shortest u, v-path,
2) the edge v, v’ within X (or Y) , and
3) the reverse of a shortest u, v’-
path.

Graph Theory
56
 By Lemma 1.2.15, such a walk must contain an odd
cycle, which contradicts our hypothesis
 Hence X and Y are independent sets. Also X∪Y = V(H),
so H is an X, Y-bipartite graph
u
v
'v
Even (or Odd)
Even (or Odd)
Odd Cycle
Because:
even (or odd) + even (or odd) = even
even + 1 = odd
Since no odd cycles, vv’ doesn’t
exist.
We have:
X and Y are independent sets

Graph Theory
57
Theorem: A graph is bipartite only if it has no
odd cycle. 1.2.18
Proof: (necessity)
 Let G be a bipartite graph.
 Every walk alternates between the two sets of a
bipartition
 So every return to the original partite set
happens after an even number of steps
 Hence G has no odd cycle

Graph Theory
58
Eulerian Circuits 1.2.24
 A graph is Eulerian if it has a closed trail
containing all edges.
 We call a closed trail a circuit when we do
not specify the first vertex but keep the list in
cyclic order.
 An Eulerian circuit or Eulerian trail in a
graph is a circuit or trail containing all the
edges.

Graph Theory
59
Even Graph, Even Vertex1.2.24
 An even graph is a graph with vertex
degrees all even.
 A vertex is odd [even] when its degree is odd
[even].

Graph Theory
60
Maximal Path1.2.24
 A maximal path in a graph G is a path P in
G that is not contained in a longer path.
– When a graph is finite, no path can extend
forever , so maximal (non-extendible) paths
exist.

Graph Theory
61
Lemma: If every vertex of graph G has degree at
least 2, then G contains a cycle. 1.2.25
Proof:
 Let P be a maximal path in G, and let u be an endpoint of P
 Since P cannot be extended, every neighbor of u must
already be a vertex of P
 Since u has degree at least 2, it has a neighbor v in V (P )
via an edge not in P
 The edge uv completes a cycle with the portion of P from v
to u
uP
Impossible
v
P
u
Must

Graph Theory
62
Theorem: A graph G is Eulerian if and only if it has at most one
nontrivial component and its vertices all have even degree. 1.2.26
Proof: (Necessity)
 Suppose that G has an Eulerian circuit C
 Each passage of C through a vertex uses two
incident edges
 And the first edge is paired with the last at the first
vertex
 Hence every vertex has even degree
In
Out
Start (The 1st
)
End (The last)

Graph Theory
63
Proof: (Necessity)
 Also, two edges can be in the same trail only
when they lie in the same component, so there
is at most one nontrivial component.
Component 1 Component 2
If more than one components,
can’t walk across the graph

Graph Theory
64
Theorem: A graph G is Eulerian if and only if it has at most
one nontrivial component and its vertices all have even
degree 1.2.26
Proof: (Sufficiency 1/3)
 Assuming that the condition holds, we obtain an
Eulerian circuit using induction on the number
of edges, m
 Basis step: m= 0. A closed trail consisting of
one vertex suffices →

Graph Theory
65
 Induction step: m>0.
– When even degrees, each vertex in the nontrivial
component of G has degree at least 2.
– By Lemma 1.2.25, the nontrivial component has a
cycle C.
– Let G’ be the graph obtained from G by deleting
E(C).
– Since C has 0 or 2 edges at each vertex, each
component of G’ is also an even graph.
– Since each component is also connected and has
fewer than m edges, we can apply the induction
hypothesis to conclude that each component of G’
has an Eulerian circuit. →

Graph Theory
66
 Induction step: m>0. (continued)
– To combine these into an Eulerian circuit of G,
we traverse C, but when a component of G’ is
entered for the first time we detour along an
Eulerian circuit of that component.
– This circuit ends at the vertex where we
began the detour. When we complete the
traversal of C, we have completed an Eulerian
circuit of G.

Graph Theory
67
Proposition: Every even graph decomposes into cycles1.2.27
Proof:
 In the proof of Theorem 1.2.26
– It is noted that every even nontrivial graph
has a cycle
– The deletion of a cycle leaves an even
graph
 Thus this proposition follows by induction on
the number of edges

Graph Theory
68
Proposition: If G is a simple graph in which every vertex has
degree at least k, then G contains a path of length at least k.
If k≥2, then G also contains a cycle of length at least k+1.
1.2.28
Proof: (1/2)
 Let u be an endpoint of a maximal path P in G.
 Since P does not extend, every neighbor of u
is in V(P).
 Since u has at least k neighbors and G is
simple, P therefore has at least k vertices
other than u and has length at least k.

Graph Theory
69
Proposition: If G is a simple graph in which every vertex has
degree at least k, then G contains a path of length at least k.
If k≥2, then G also contains a cycle of length at least k+1.
1.2.28
Proof: (2/2)
 If k ≥ 2, then the edge from u to its farthest
neighbor v along P completes a sufficiently
long cycle with the portion of P from v to u.
u
v
d(u) ≥ k
At least k+1 vertices
Length ≥ k

Graph Theory
70
Degree1.3.1
 The degree of vertex v in a graph G, written
or d (v ), is the number of edges incident
to v, except that each loop at v counts twice
 The maximal degree is ∆(G )
 The minimum degree is δ (G )
A
C
B
D
F
E
d(B) = 3, d(C) = 2
Δ(G) = 3, δ(G) = 2
G

Graph Theory
71
Regular 1.3.1
 G is regular if ∆(G ) = δ (G )
 G is k-regular if the common degree is k.
 The neighborhood of v, written Ng (v ) or N (v )
is the set of vertices adjacent to v.
3-regular

Graph Theory
72
Order and size 1.3.2
 The order of a graph G, written n (G ), is the
number of vertices in G.
 An n-vertex graph is a graph of order n.
 The size of a graph G, written e (G ), is the
number of edges in G.
 For n∈N, the notation [n ] indicates the set
{1,…, n }.

Graph Theory
73
Proposition: (Degree-Sum Formula)
If G is a graph, then Σv∈V(G)d(v) = 2e(G) 1.3.3
Proof:
 Summing the degrees counts each edge twice,
– Because each edge has two ends and
contributes to the degree at each endpoint.

Graph Theory
74
Theorem: If k>0, then a k-regular bipartite graph has
the same number of vertices in each partite set. 1.3.9
Proof:
 Let G be an X,Y - bigraph.
 Counting the edges according to their
endpoints in X yields e (G ) = k |X |.
d (x) = k
x

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75
Theorem: If k>0, then a k-regular bipartite graph has
the same number of vertices in each partite set. 1.3.9
Proof:
 Counting them by their endpoints in Y yields
e (G )=k |Y |
 Thus k |X | = k |Y |, which yields |X |=|Y | when
k > 0
d (x) = k
x
d (y) = ky

Graph Theory
76
A technique for counting a set 1/3 1.3.10
 Example: The Petersen graph has ten 6-cycles
– Let G be the Petersen graph.
– Being 3-regular, G has ten copies of K1,3(claw). We
establish a one-to-one correspondence between the
6-cycles and the claws.
– Since G has girth 5, every 6-cycle F is an induced
subgraph.
• see below
– Each vertex of F has one neighbor outside F.
• d(v)= 3, v ∈V(G)
If Existing, Girth =3.
But Girth=5 so no such an edge

Graph Theory
77
A technique for counting a set 2/3 1.3.10
– Since nonadjacent vertices have exactly one
common neighbor (Proposition 1.1.38), opposite
vertices on F have a common neighbor outside F.
– Since G is 3-regular, the resulting three vertices
outside F are distinct.
– Thus deleting V(F) leaves a subgraph with three
vertices of degree 1 and one vertex of degree 3; it
is a claw.
Common neighbor
of opposite vertices
If the neighbors are
not distinct, d(v)>3

Graph Theory
78
A technique for counting a set 3/3 3.10
– It is shown that each claw H in G arises exactly
once in this way.
– Let S be the set of vertices with degree 1 in H; S is
an independent set.
– The central vertex of H is already a common
neighbor, so the six other edges from S reach
distinct vertices.
– Thus G-V(H) is 2-regular. Since G has girth 5, G-
V(H) must be a 6-cycle. This 6-cycle yields H when
its vertices are deleted.

Graph Theory
79
Proposition: The minimum number of edges in a connected
graph with n vertices is n-1. 3.13
Proof:
 By proposition 1.2.11, every graph with n
vertices and k edges has at least n-k
components.
 Hence every n-vertex graph with fewer than n-1
edges has at least two components and is
disconnected.
 The contrapositive of this is that every
connected n-vertex graph has at least n-1
edges. This lower bound is achieved by the
path Pn.

Graph Theory
80
Theorem: If G is simple n-vertex graph with
δ(G)≥(n-1)/2, then G is connected. 1.3.15
Proof: 1/2
 Choose u,v ∈ V (G ).
 It suffices to show that u,v have a common
neighbor if they are not adjacent.
 Since G is simple, we have
|N(u) | ≥ δ (G ) ≥ (n-1)/2,
and similarly for v.
– Recall: δ (G ) is the minimum degree,
|N(u)| = d(u) Hence: |N(u) | ≥ δ (G )

Graph Theory
81
Theorem: If G is simple n-vertex graph with
δ(G)≥(n-1)/2, then G is connected. 1.3.15
Proof: 2/2
 When u and v are not connected, we have |N(u
) ∪N(v )| ≤ n - 2
– since u and v are not in the union
 Using Remark A.13 of Appendix A, we thus
compute
| ( ) ( )| | ( )| | ( )| | ( ) ( )|
1 1 ( 2) 1.
2 2
N u N v N u N v N u N v
n n n
∩ = + − ∪
− −≥ + − − =

Graph Theory
82
Theorem: Every loopless graph G has a bipartite
subgraph with at least e(G)/2 edges. 1.3.19
Proof:
 Partition V(G) into two sets X, Y.
 Using the edges having one endpoint in each set yields
a bipartite subgraph H with bipartition X, Y.
 If H contains fewer than half the edges of G incident to
a vertex v, then v has more edges to vertices in its own
class than in the other class, as illustrated bellow.

Graph Theory
83
Proof: 2/2
 Moving v to the other class gains more edges of
G than it loses.
 Using Iterative improvement approach
 When it terminates, we have dH(v) ≥ dG(v)/2 for
every v∈V(G) .
 Summing this and applying the degree-sum
formula yields e(H) ≥ e(G)/2.
Theorem: Every loopless graph G has a bipartite
subgraph with at least e(G)/2 edges. 1.3.19

Graph Theory
84
Example1 1.3.20
 The algorithm in Theorem 1.3.19 need not
produce a bipartite subgraph with the most
edges, merely one with at least half the edges
– Local Maximum

Graph Theory
85
Example2 1.3.20
 Consider the graph in the next page.
– It is 5-regular with 8 vertices and hence has
20 edges.
– The bipartition X={a,b,c,d} and Y={e,f,g,h}
yields a 3-regular bipartite subgraph with 12
edges.
– The algorithm terminates here:
• switching one vertex would pick up two
edges but lose three .

Graph Theory
86
Example(Cont.) 1.3.20
a
b
c
de
f
g
h switching a would pick
up two edges but lose
three

Graph Theory
87
Example 1.3.20
 Nevertheless, the bipartition X={a,b,g,h} and
Y={c,d,e,f} yields a 4-regular bipartite subgraph
with 16 edges.
 An algorithm seeking the maximal by local
changes may get stuck in a local maximum.
a
b
c
de
f
g
h a
b
c
de
f
g
hLocal
Maximum
Global
Maximum

Graph Theory
88
Theorem: The maximum number of edges in an n-
vertex triangle free simple graph is  n2
/4  1.3.23
Proof : 1/6
– Let G be an n-vertex triangle-free simple graph.
– Let x be a vertex of maximum degree and d(x)=k.
– Since G has no triangles, there are no edges
among neighbors of x.
No edges between
neighbors of x

Graph Theory
89
/4  1.3.23
Proof : 2/6
– Hence summing the degrees of x and its
nonneighbors counts at least one endpoint
of every edge: Σv∉N(x)d(v) ≥ e(G).
– We sum over n-k vertices, each having
degree at most k, so e(G) ≤ (n-k)k

Graph Theory
90
)(xN
x x
)(xN
Doesn’t exist
• Σv∉N(x) d(v) counts at least
one endpoint of every edge
At most k vertices
At least n-k vertices
No edges exist
/4  1.3.23
Proof: 3/6

Graph Theory
91
Proof: 4/6
– Since (n-k)k counts the edges in Kn-k, k, we have
now proved that e(G) is bounded by the size
of some biclique with n vertices.
• i.e. e(G) ≤ (n-k)k = |the edges in Kn-k,k|
/4  1.3.23
n-k k

Graph Theory
92
Proof: 5/6
– Moving a vertex of Kn-k,k from the set of size k
to the set of size n-k gains k-1 edges and
loses n-k edges.
– The net gain is 2k-1-n, which is positive for
2k>n+1 and negative for 2k<n+1.
– Thus e(Kn-k, k) is maximized when k is n/2  or
n/2 .
/4  1.3.23
n-k k

Graph Theory
93
Proof: 6/6
– The product is then n2
/4 for even n and (n2
-1)/4
for odd n. Thus e(G) ≤ n2
/4 .
– The bound is best possible.
• It is seen that a triangle-free graph with n2
/4
edges is: Kn/2,n/2.
/4  1.3.23

Graph Theory
94
Degree sequence 1.3.27
 The Degree Sequence of a graph is the list of
vertex degrees, usually written in non-
increasing order, as d1≥ …. ≥ dn .
 Example:
z
y
x
w
v
Degree sequence:
d(w), d(x), d(y), d(z), d(v)

Graph Theory
95
Proposition: The nonnegative integers d1 ,…, dn are the
vertex degrees of some graph if and only if ∑di is even.
1.3.28
Proof: ½ Necessity
 When some graph G has these numbers as its
vertex degrees, the degree-sum formula implies
that ∑di = 2e (G), which is even.

Graph Theory
96
Proof: 2/2 Sufficiency
 Suppose that ∑ di is even.
 We construct a graph with vertex set v1,…,vn and d(vi) = di
for all i.
 Since ∑ di is even, the number of odd values is even.
 First form an arbitrary pairing of the vertices in {vi : di is
odd}.
 For each resulting pair, form an edge having these two
vertices as its endpoints
 The remaining degree needed at each vertex is even and
nonnegative; satisfy this for each i by placing [di /2] loops
at vi
Proposition: The nonnegative integers d1 ,…, dn are the
vertex degrees of some graph if and only if ∑di is even.
1.3.28

Graph Theory
97
Graphic Sequence 1.3.29
 A graphic sequence is a list of nonnegative
numbers that is the degree sequence of
some simple graph.
 A simple graph “realizes” d.
– means: A simple graph with degree
sequence d.

Graph Theory
98
Recursive condition 1.3.30
 The lists (2, 2, 1, 1) and (1, 0, 1) are graphic. The graphic
K2+K1 realizes 1, 0, 1.
 Adding a new vertex adjacent to vertices of degrees 1
and 0 yields a graph with degree sequence 2, 2, 1, 1, as
shown below.
 Conversely, if a graph realizing 2, 2, 1, 1 has a vertex w
with neighbors of degrees 2 and 1, then deleting w
yields a graph with degrees 1, 0, 1.
w
K2
K1
1
1
0
1 1
2
2

Graph Theory
99
 Similarly, to test 33333221, we seek a
realization with a vertex w of degree 3 having
three neighbors of degree 3.
3 3 3 3 3 2 2 1
2 2 2 3 2 2 1
Delete this
Vertex
A new
degree sequence

Graph Theory
100
This exists if and only if 2223221 is graphic. (See next page)
– We reorder this and test 3222221.
– We continue deleting and reordering until we can
tell whether the remaining list is realizable.
– If it is, then we insert vertices with the desired
neighbors to walk back to a realization of the
original list.
– The realization is not unique.
 The next theorem implies that this recursive test works.

Graph Theory
101
33333221 3222221 221111
11100
2223221 111221 10111
wv
u u
v
u

Graph Theory
102
Theorem. For n>1, an integer list d of size n is graphic if and
only if d’ is graphic, where d’ is obtained from d by deleting
its largest element ∆ and subtracting 1 from its ∆ next largest
elements. The only 1-element graphic sequence is d1=0. 1.3.31
Proof: 1/6
 For n =1, the statement is trivial.
 For n >1, we first prove that the condition is sufficient.
– Give d with d1≥…..≥dn and a simple graph G’ with
degree sequence d’
For Example:
We have: 1) d = 33333221
2) G’ with d’ = 2223221
We show: d is graphic
G’

Graph Theory
103
Theorem. For n>1, an integer list d of size n is graphic if and
only if d’ is graphic, where d’ is obtained from d by deleting
its largest element ∆ and subtracting 1 from its ∆ next largest
elements. The only 1-element graphic sequence is d1=0. 1.3.31
Proof: 2/6
– We add a new vertex adjacent to vertices in G’ with
degrees d2-1,…..,d∆+1-1.
– These di are the ∆ largest elements of d after (one
copy of) ∆ itself,
– Note : d2-1,…..,d∆+1-1 need not be the ∆ largest
numbers in d’ (see example in previous page)
G’
New added
vertex
d : d1,d2,… dn
d’ : d2-1,…..,d∆+1-1,… dn
May not be the ∆
largest numbers

Graph Theory
104
Theorem 1.3.31 continue
 To prove necessity, 3/6
– Given a simple graph G realizing d , we produce a
simple graph G’ realizing d’
– Let w be a vertex of degree ∆ in G, and let S be a
set of ∆ vertices in G having the “desired degrees”
d2,…..,d∆+1
d : d1, d2, …d∆, d∆+1,… dn
S: ∆ verticesd1=∆
w

Graph Theory
105
Theorem 1.3.31
Proof: continue 4/6
– If N(w)=S, then we delete w to obtain G’.
d : d1, d2, …d∆, d∆+1,… dn
∆Vertices, N(w)=S
i.e. They are connected to w
d1=∆
w
Delete w than we have
d’ : d2-1,…..,d∆+1-1,… dn

Graph Theory
106
Theorem 1.3.31
Proof: continue 5/6
– Otherwise,
• Some vertex of S is missing from N(w).
• In this case, we modify G to increase |N(w)∩S| without
changing any vertex degree.
• Since |N(w)∩S| can increase at most ∆ times, repeating
this converts G into another graph G* that realizes d and
has S as the neighborhood of w.
• From G* we then delete w to obtain the desired graph
G’ realizing d’.
d : d1, d2, …d∆, d∆+1,… dn
∆Vertices, N(w)≠S
i.e. Some vertices are not connected to w.
- We make them become connected to w
without changing their degree.
d1=∆
w

Graph Theory
Theorem 1.3.31
Proof: continue 6/6
• To find the modification when N(w)≠S , we choose
x∈S and z∉S so that w z are connected and w x are
not.
• We want to add wx and delete wz, but we must
preserve vertex degrees. Since d(x)>d(z) and already
w is a neighbor of z but not x, there must be a vertex y
adjacent to x but not to z. Now we delete {wz,xy} and
add {wx,yz} to increase |N(w)∩S| .
w
z
x
y
This y must exist.
w
z
x
y⇒
It becomes connected w

Graph Theory
108
2-switch 1.3.32
 A 2-switch is the replacement of a pair of
edges xy and zw in a simple graph by the
edges yz and wx, given that yz and wx did not
appear in the graph originally.
wx
y z y z
wx

Graph Theory
109
Theorem: If G and H are two simple graphs with vertex set V,
then dG(v)=dH(v) for every v∈V if and only if there is a
sequence of 2-switches that transforms G into H. 1.3.33
Proof:
 Every 2-switch preserves vertex degrees, so
the condition is sufficient.
 Conversely, when dG(v)=dH(v) for all v∈V , we
obtain an appropriate sequence of 2-switches
by induction on the number of vertices, n.
 If n<3, then for each d1,…..,dn there is at most
one simple graph with d(vi)=di.
 Hence we can use n=3 as the basis step.

Graph Theory
110
Theorem. 1.3.33 (Continue)
 Consider n≥4 , and let w be a vertex of maximum
degree,∆
 Let S={v1,…..,v∆} be a fixed set of vertices with the ∆
highest degrees other than w
 As in the proof of Theorem 1.3.31, some sequence
of 2-switches transforms G to a graph G* such that
NG*(w)=S, and some such sequence transforms H to
a graph H* such that NH*(w)=S
 Since NG*(w)=NH*(w), deleting w leaves simple graphs
G’=G*-w and H’=H*-w with dG’(v)=dH’(v) for every
vertex v

Graph Theory
111
Theorem. 1.3.33 Continue
 By the induction hypothesis, some sequence of 2-
switches transforms G’ to H’. Since these do not
involve w, and w has the same neighbors in G* and
H*, applying this sequence transforms G* to H*.
 Hence we can transform G to H by transforming G to
G*, then G* to H*, then (in reverse order) the
transformation of H to H*.

Graph Theory
112
Directed Graph and Its edges 1.4.2
 A directed graph or digraph G is a triple:
– A vertex set V(G),
– An edge set E(G), and
– A function assigning each edge an ordered pair of
vertices.
• The first vertex of the ordered pair is the tail of the
edge
• The second is the head
• Together, they are the endpoints.
 An edge is said to be from its tail to its head.
– The terms “head” and “tail” come from the arrows
used to draw digraphs.

Graph Theory
113
Directed Graph and its edges 1.4.2
 As with graphs, we
– assign each vertex a point in the plane and
– each edge a curve joining its endpoints.
 When drawing a digraph, we give the curve a direction
from the tail to the head.

Graph Theory
114
Directed Graph and its edges 1.4.2
 When a digraph models a relation, each ordered pair is
the (head, tail) pair for at most one edge.
– In this setting as with simple graphs, we ignore the
technicality of a function assigning endpoints to edges
and simply treat an edge as an ordered pair of
vertices.

Graph Theory
115
Loop and multiple edges in directed graph 1.4.3
 In a graph, a loop is an edge whose endpoints are
equal.
 Multiple edges are edges having the same ordered
pair of endpoints.
 A digraph is simple if each ordered pair is the head
and tail of the most one edge; one loop may be present
at each vertex.
Loop
Multiple
edges

Graph Theory
116
Loop and multiple edges in directed graph 1.4.3
 In the simple digraph, we write uv for an edge
with tail u and head v.
– If there is an edge form u to v, then v is a
successor of u, and u is a predecessor of
v.
– We write u→v for “there is an edge from u to
v”.
Predecessor
Successor

Graph Theory
117
Path and Cycle in Digraph 1.4.6
 A digraph is a path if it is a simple digraph
whose vertices can be linearly ordered so that
there is an edge with tail u and head v if and
only if v immediately follows u in the vertex
ordering.
 A cycle is defined similarly using an ordering
of the vertices on the cycle.

Graph Theory
118
Underlying graph 1.4.9
 The underlying graph of a digraph D:
– the graph G obtained by treating the edges of
D as unordered pairs;
– the vertex set and edges set remain the
same, and the endpoints of an edge are the
same in G as in D,
– but in G they become an unordered pair.
The underlying GraphA digraph

Graph Theory
119
Underlying graph 1.4.9
 Most ideals and methods of graph theorem
arise in the study of ordinary graphs.
 Digraphs can be a useful additional tool,
especially in applications
 When comparing a digraph with a graph, we
usually use G for the graph and D for the
digraph. When discussing a single digraph, we
often use G.

Graph Theory
120
Adjacency Matrix and Incidence Matrix
of a Digraph 1.4.10
 In the adjacency matrix A(G) of a digraph
G, the entry in position i, j is the number of
edges from vi to vj.
 In the incidence matrix M(G) of a loopless
digraph G, we set mi,j=+1 if viis the tail of ejand
mi,j= -1 if viis the head of ej.

Graph Theory
121
Example of adjacency matrix 1.4.11
 The underlying graph of the digraph below is the
graph of Example 1.1.19; note the similarities and
differences in their matrices.












0000
1010
0101
0100












−
++−−
−++
+−
10000
11110
01101
00011
w x y z
w
x
y
z
w
x
y
z
a b c d e
)(GA G )(GM
a
b
ec
d
w
x
y z

Graph Theory
122
Connected Digraph 1.4.12
 To define connected digraphs, two options
come to mind. We could require only that
the underlying graph be connected.
 However, this does not capture the most
useful sense of connection for digraphs.

Graph Theory
123
Weakly and strongly connected digraphs 1.4.12
 A graph is weakly connected if its
underlying graph is connected.
 A digraph is strongly connected or
strong if for each ordered pair u,v of vertices,
there is a path from u to v.

Graph Theory
124
Eulerian Digraph 1.4.22
 An Eulerian trail in digraph (or graph) is a
trail containing all edges.
 An Eulerian circuit is a closed trail
containing all edges.
 A digraph is Eulerian if it has an Eulerian
circuit.

Graph Theory
125
Lemma. If G is a digraph with δ+
(G)≥1, then G
contains a cycle. The same conclusion holds
when δ-
(G) ≥1. 1.4.23
Proof.
 Let P be a maximal path in G, and u be the last
vertex of P.
 Since P cannot be extended, every successor of
u must already be a vertex of P.
 Since δ+
(G)≥1, u has a successor v on P.
 The edge uv completes a cycle with the portion
of P from v to u.

Graph Theory
126
Theorem: A digraph is Eulerian if and only if
d+
(v)=d-
(v) for each vertex v and the
underlying graph has at most one nontrivial
component. 1.4.24

Graph Theory
127
De Bruijn cycles 1.4.25
 Application:
– There are 2n
binary strings of length n.
– Is there a cyclic arrangement of 2n
binary digits such
that the 2n
strings of n consecutive digitals are all
distinct?
– Example: For n =4, (0000111101100101) works.
0000 0001 0011 0111 1111 1110 1101 1011 …
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1

Graph Theory
128
 We can use such an arrangement to keep track of the
position of a rotating drum.
– One drum has 2n
rotational positions.
– A band around the circumference is split into 2n
portions that can be coded 0 or 1.
– Sensors read n consecutive portions.
– If the coding has the property specified above, then
the position of the drum is determined by the string
read by the sensors.

Graph Theory
129
 To obtain such a circular arrangement,
– define a digraph Dnwhose vertices are the
binary (n-1)-tuples.
– Put an edge from a to b if the last n-2 entries
of a agree with the first n-2 entries of b.
– Label the edge with the last entry of b.

Graph Theory
130
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
0 0
0
0
0
0
0
1
1
1
1
1 1
1
1
001
000
100
010 101
011
110
111
a b
Put an edge from a to b if the last n-2
entries of a agree with the first n-2
entries of b.
Label the edge with the last entry of
b.
 Below we show D4..

Graph Theory
131
 We next prove that Dn is Eulerian and show how
an Eulerian circuit yields the desired circular
arrangement.

Graph Theory
132
Theorem. The digraph Dn of Application 1.4.25 is Eulerian, and
the edge labels on the edges in any Eulerian circuit of Dn form
a cyclic arrangement in which the 2n
consecutive segments of
length n are distinct. 1.4.26
Proof:
 We show
– first that Dn is Eulerian.
– Then the labels on the edges in any
Eulerian circuit of Dn form a cyclic
arrangement in which the 2n
consecutive
segments of length n are distinct.

Graph Theory
133
Theorem. The digraph Dn is Eulerian 1.4.26
Proof: 1/2
 Every vertex has out-degree 2
– because we can append a 0 or a 1 to its name to obtain
the name of a successor vertex.
 Similarly, every vertex has in-degree 2,
– because the same argument applies when moving in
reverse and putting a 0 or a 1 on the front of the name.
001
101
011
110
111

Graph Theory
134
Theorem. The digraph Dn is Eulerian 1.4.26
Proof: 2/2
 Also, Dn is strongly connected,
– because we can reach the vertex b=(b1,…..,bn-1)
from any vertex by successively follows the
edges labeled b1,…..,bn-1.
 Thus Dn satisfies the hypotheses of Theorem
1.4.24 and is Eulerian.
0
0 0
0
0
0
0
0
1
1
1
1 1
1
1
001
000
100
010 101
011
110
111
a b1
1

Graph Theory
135
Theorem. The labels on the edges in any Eulerian circuit of Dn
form a cyclic arrangement in which the 2n
consecutive
segments of length n are distinct. 1.4.26
Proof: 1/4
 Let C be an Eulerian circuit of Dn. Arrival at vertex
a=(a1,…..,an-1) must be along an edge with label an-1
– because the label on an edge entering a vertex
agrees with the last entry of the name of the
vertex.
0
0 0
0
0
0
0
0
1
1
1
1
1 1
1
1
001
000
100
010 101
011
110
111
a b

Graph Theory
136
consecutive
Proof: 2/4
 The successive earlier labels (looking backward)
must have been an-2,…..,a1 in order.
– because we delete the front and shift the reset
to obtain the reset of the name at the head
0
0 0
0
0
0
0
0
1
1
1
1
1 1
1
1
001
000
100
010 101
0 1 1
110
111
a b

Graph Theory
137
consecutive
Proof: 2/4
 If C next uses an edge with label an, then the list
consisting of the n most recent edge labels at that
time is a1,…..an.
0
0
1
0
0
0
0
0
1
1
1
1
1 1
1
1
001
000
100
010 101
011
110
111
a b
0
1

Graph Theory
138
Theorem. The labels on the edges in any Eulerian circuit of
Dn form a cyclic arrangement in which the 2n
consecutive
Proof: 3/4
 Since
– the 2n-1
vertex labels are distinct, and
– the two out-going edges have distinct labels,
and
– we traverse each edge exactly once
011 0
011 1
Distinct
vertex label
Distinct labels on
out-going edges

Graph Theory
139
Theorem. The labels on the edges in any Eulerian circuit of
Dn form a cyclic arrangement in which the 2n
consecutive
Proof: 4/4
 We have shown that the 2n
strings of length n in
the circular arrangement given by the edge
labels along C are distinct.

Graph theory

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