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Graph Theory
Ch. 1. Fundamental Concept
1
Chapter 1
Fundamental Concept
1.1 What Is a Graph?
1.2 Paths, Cycles, and Trails...
Graph Theory
Ch. 1. Fundamental Concept
2
The KÖnigsberg Bridge Problem
 Königsber is a city on the Pregel river in Pruss...
Graph Theory
Ch. 1. Fundamental Concept
3
A Model
 A vertex : a region
 An edge : a path(bridge) between two
regions
e1
...
Graph Theory
Ch. 1. Fundamental Concept
4
General Model
 A vertex : an object
 An edge : a relation between two objects
...
Graph Theory
Ch. 1. Fundamental Concept
5
What Is a Graph?
 A graph G is a triple consisting of:
– A vertex set V(G )
– A...
Graph Theory
Ch. 1. Fundamental Concept
6
Loop, Multiple edges
 Loop : An edge whose endpoints are equal
 Multiple edges...
Graph Theory
Ch. 1. Fundamental Concept
7
Simple Graph
 Simple graph : A graph has no loops or multiple
edges
loop
Multip...
Graph Theory
Ch. 1. Fundamental Concept
8
Adjacent, neighbors
 Two vertices are adjacent and are neighbors
if they are th...
Graph Theory
Ch. 1. Fundamental Concept
9
Finite Graph, Null Graph
 Finite graph : an graph whose vertex set and
edge set...
Graph Theory
Ch. 1. Fundamental Concept
10
Complement
 Complement of G: The complement G’ of a
simple graph G :
– A simpl...
Graph Theory
Ch. 1. Fundamental Concept
11
Clique and Independent set
 A Clique in a graph: a set of pairwise
adjacent ve...
Graph Theory
Ch. 1. Fundamental Concept
12
Bipartite Graphs
 A graph G is bipartite if V(G) is the union of
two disjoint ...
Graph Theory
Ch. 1. Fundamental Concept
13
Chromatic Number
 The chromatic number of a graph G,
written x(G), is the mini...
Graph Theory
Ch. 1. Fundamental Concept
14
Maps and coloring
 A map is a partition of the plane into
connected regions
 ...
Graph Theory
Ch. 1. Fundamental Concept
15
Scheduling and graph Coloring 1
 Two committees can not hold meetings at the
s...
Graph Theory
Ch. 1. Fundamental Concept
16
Scheduling and graph Coloring 1
 Model:
– One committee being represented by a...
Graph Theory
Ch. 1. Fundamental Concept
17
Scheduling and graph Coloring 2
 Scheduling problem is equivalent to graph
col...
Graph Theory
Ch. 1. Fundamental Concept
18
Path and Cycle
 Path : a sequence of distinct vertices such
that two consecuti...
Graph Theory
Ch. 1. Fundamental Concept
19
Subgraphs
 A subgraph of a graph G is a graph H such
that:
– V(H) ⊆ V(G) and E...
Graph Theory
Ch. 1. Fundamental Concept
20
Subgraphs
 Example: H1, H2, and H3 are subgraphs of G
c
d
a b
de
a b
c
de
H1
G...
Graph Theory
Ch. 1. Fundamental Concept
21
Connected and Disconnected
 Connected : There exists at least one path
between...
Graph Theory
Ch. 1. Fundamental Concept
22
Adjacency, Incidence, and Degree
 Assume ei is an edge whose endpoints are
(vj...
Graph Theory
Ch. 1. Fundamental Concept
23
Adjacency matrix
 Let G = (V, E), |V| = n and |E|=m
 The adjacency matrix of ...
Graph Theory
Ch. 1. Fundamental Concept
24
Incidence Matrix
 Let G = (V, E), |V| = n and |E|=m
 The incidence matrix M(G...
Graph Theory
Ch. 1. Fundamental Concept
25
Isomorphism
 An isomorphism from a simple graph G to a
simple graph H is a bij...
Graph Theory
Ch. 1. Fundamental Concept
26
Complete Graph
 Complete Graph : a simple graph whose
vertices are pairwise ad...
Graph Theory
Ch. 1. Fundamental Concept
27
Complete Bipartite Graph or Biclique
 Complete bipartite graph (biclique) is a...
Graph Theory
Ch. 1. Fundamental Concept
28
Petersen Graph 1.1.36
 The petersen graph is the simple graph
whose vertices a...
Graph Theory
Ch. 1. Fundamental Concept
29
Petersen Graph 1.1.37
 Assume: the set of 5-element be (1, 2, 3, 4, 5)
– Then,...
Graph Theory
Ch. 1. Fundamental Concept
30
Petersen Graph 1.1.36
 Three drawings
Graph Theory
Ch. 1. Fundamental Concept
31
Theorem: If two vertices are non-adjacent in the
Petersen Graph, then they have...
Graph Theory
Ch. 1. Fundamental Concept
32
Girth 1.1.39, 1.1.40
 Girth : the length of its shortest cycle.
– If no cycles...
Graph Theory
Ch. 1. Fundamental Concept
33
Girth and Petersen graph 1.1.39, 1.1.40
 Theorem: The Petersen Graph has girth...
Graph Theory
Ch. 1. Fundamental Concept
34
Walks, Trails1.2.2
 A walk : a list of vertices and edges v0, e1, v1,
…., ek, ...
Graph Theory
Ch. 1. Fundamental Concept
35
Paths 1.2.2
 A u,v-walk or u,v-trail has first vertex u and last
vertex v; the...
Graph Theory
Ch. 1. Fundamental Concept
36
Lemma: Every u,v-walk contains a u,v-path 1.2.5
Proof:
 Use induction on the l...
Graph Theory
Ch. 1. Fundamental Concept
37
Lemma: Every u,v-walk contains a u,v-path 1.2.5
Proof: Continue
 Induction ste...
Graph Theory
Ch. 1. Fundamental Concept
38
Lemma: Every u,v-walk contains a u,v-path 1.2.5
Proof: Continue
 Induction ste...
Graph Theory
Ch. 1. Fundamental Concept
39
Components 1.2.8
 The components of a graph G are its
maximal connected subgra...
Graph Theory
Ch. 1. Fundamental Concept
40
Proof:
– An n-vertex graph with no edges has n
components
– Each edge added red...
Graph Theory
Ch. 1. Fundamental Concept
41
Theorem: Every graph with n vertices and k edges
has at least n-k components 1....
Graph Theory
Ch. 1. Fundamental Concept
42
Cut-edge, Cut-vertex 1.2.12
 A cut-edge or cut-vertex of a graph is an
edge or...
Graph Theory
Ch. 1. Fundamental Concept
43
Cut-edge, Cut-vertex 1.2.12
 G-e or G-M : The subgraph obtained by
deleting an...
Graph Theory
Ch. 1. Fundamental Concept
44
Induced subgraph 1.2.12
 An induced subgraph :
– A subgraph obtained by deleti...
Graph Theory
Ch. 1. Fundamental Concept
45
Induced subgraph 1.2.12
 More Examples:
– G2 is the subgraph of G1 induced by
...
Graph Theory
Ch. 1. Fundamental Concept
46
Induced subgraph 1.2.12
 A set S of vertices is an independent set if and
only...
Graph Theory
Ch. 1. Fundamental Concept
47
Theorem: An edge e is a cut-edge if and only if e
belongs to no cycles. 1.2.14
...
Graph Theory
Ch. 1. Fundamental Concept
48
Theorem: An edge e is a cut-edge if and only if e
belongs to no cycles. 1.2.14
...
Graph Theory
Ch. 1. Fundamental Concept
49
Theorem: An edge e is a cut-edge if and only
if e belongs to no cycles. 1.2.14
...
Graph Theory
Ch. 1. Fundamental Concept
50
Lemma: Every closed odd walk contains an odd cycle
Proof:1/3
 Use induction on...
Graph Theory
Ch. 1. Fundamental Concept
51
Lemma: Every closed odd walk contains an odd cycle
Proof: 2/3
 Suppose that th...
Graph Theory
Ch. 1. Fundamental Concept
52
Lemma: Every closed odd walk contains an odd cycle
Proof: 3/3
– If W has a repe...
Graph Theory
Ch. 1. Fundamental Concept
53
Theorem: A graph is bipartite if and only if it has
no odd cycle. 1.2.18
 Exam...
Graph Theory
Ch. 1. Fundamental Concept
54
Theorem: A graph is bipartite if it has no odd cycle. 1.2.18
Proof: (sufficienc...
Graph Theory
Ch. 1. Fundamental Concept
55
Theorem: A graph is bipartite if it has no odd cycle. 1.2.18
Proof: (sufficienc...
Graph Theory
Ch. 1. Fundamental Concept
56
Theorem: A graph is bipartite if it has no odd cycle. 1.2.18
Proof: (sufficienc...
Graph Theory
Ch. 1. Fundamental Concept
57
Theorem: A graph is bipartite only if it has no
odd cycle. 1.2.18
Proof: (neces...
Graph Theory
Ch. 1. Fundamental Concept
58
Eulerian Circuits 1.2.24
 A graph is Eulerian if it has a closed trail
contain...
Graph Theory
Ch. 1. Fundamental Concept
59
Even Graph, Even Vertex1.2.24
 An even graph is a graph with vertex
degrees al...
Graph Theory
Ch. 1. Fundamental Concept
60
Maximal Path1.2.24
 A maximal path in a graph G is a path P in
G that is not c...
Graph Theory
Ch. 1. Fundamental Concept
61
Lemma: If every vertex of graph G has degree at
least 2, then G contains a cycl...
Graph Theory
Ch. 1. Fundamental Concept
62
Theorem: A graph G is Eulerian if and only if it has at most one
nontrivial com...
Graph Theory
Ch. 1. Fundamental Concept
63
Theorem: A graph G is Eulerian if and only if it has at most one
nontrivial com...
Graph Theory
Ch. 1. Fundamental Concept
64
Theorem: A graph G is Eulerian if and only if it has at most
one nontrivial com...
Graph Theory
Ch. 1. Fundamental Concept
65
Theorem: A graph G is Eulerian if and only if it has at most one
nontrivial com...
Graph Theory
Ch. 1. Fundamental Concept
66
Theorem: A graph G is Eulerian if and only if it has at most one
nontrivial com...
Graph Theory
Ch. 1. Fundamental Concept
67
Proposition: Every even graph decomposes into cycles1.2.27
Proof:
 In the proo...
Graph Theory
Ch. 1. Fundamental Concept
68
Proposition: If G is a simple graph in which every vertex has
degree at least k...
Graph Theory
Ch. 1. Fundamental Concept
69
Proposition: If G is a simple graph in which every vertex has
degree at least k...
Graph Theory
Ch. 1. Fundamental Concept
70
Degree1.3.1
 The degree of vertex v in a graph G, written
or d (v ), is the nu...
Graph Theory
Ch. 1. Fundamental Concept
71
Regular 1.3.1
 G is regular if ∆(G ) = δ (G )
 G is k-regular if the common d...
Graph Theory
Ch. 1. Fundamental Concept
72
Order and size 1.3.2
 The order of a graph G, written n (G ), is the
number of...
Graph Theory
Ch. 1. Fundamental Concept
73
Proposition: (Degree-Sum Formula)
If G is a graph, then Σv∈V(G)d(v) = 2e(G) 1.3...
Graph Theory
Ch. 1. Fundamental Concept
74
Theorem: If k>0, then a k-regular bipartite graph has
the same number of vertic...
Graph Theory
Ch. 1. Fundamental Concept
75
Theorem: If k>0, then a k-regular bipartite graph has
the same number of vertic...
Graph Theory
Ch. 1. Fundamental Concept
76
A technique for counting a set 1/3 1.3.10
 Example: The Petersen graph has ten...
Graph Theory
Ch. 1. Fundamental Concept
77
A technique for counting a set 2/3 1.3.10
– Since nonadjacent vertices have exa...
Graph Theory
Ch. 1. Fundamental Concept
78
A technique for counting a set 3/3 3.10
– It is shown that each claw H in G ari...
Graph Theory
Ch. 1. Fundamental Concept
79
Proposition: The minimum number of edges in a connected
graph with n vertices i...
Graph Theory
Ch. 1. Fundamental Concept
80
Theorem: If G is simple n-vertex graph with
δ(G)≥(n-1)/2, then G is connected. ...
Graph Theory
Ch. 1. Fundamental Concept
81
Theorem: If G is simple n-vertex graph with
δ(G)≥(n-1)/2, then G is connected. ...
Graph Theory
Ch. 1. Fundamental Concept
82
Theorem: Every loopless graph G has a bipartite
subgraph with at least e(G)/2 e...
Graph Theory
Ch. 1. Fundamental Concept
83
Proof: 2/2
 Moving v to the other class gains more edges of
G than it loses.
...
Graph Theory
Ch. 1. Fundamental Concept
84
Example1 1.3.20
 The algorithm in Theorem 1.3.19 need not
produce a bipartite ...
Graph Theory
Ch. 1. Fundamental Concept
85
Example2 1.3.20
 Consider the graph in the next page.
– It is 5-regular with 8...
Graph Theory
Ch. 1. Fundamental Concept
86
Example(Cont.) 1.3.20
a
b
c
de
f
g
h switching a would pick
up two edges but lo...
Graph Theory
Ch. 1. Fundamental Concept
87
Example 1.3.20
 Nevertheless, the bipartition X={a,b,g,h} and
Y={c,d,e,f} yiel...
Graph Theory
Ch. 1. Fundamental Concept
88
Theorem: The maximum number of edges in an n-
vertex triangle free simple graph...
Graph Theory
Ch. 1. Fundamental Concept
89
Theorem: The maximum number of edges in an n-
vertex triangle free simple graph...
Graph Theory
Ch. 1. Fundamental Concept
90
)(xN
x x
)(xN
Doesn’t exist
• Σv∉N(x) d(v) counts at least
one endpoint of ever...
Graph Theory
Ch. 1. Fundamental Concept
91
Proof: 4/6
– Since (n-k)k counts the edges in Kn-k, k, we have
now proved that ...
Graph Theory
Ch. 1. Fundamental Concept
92
Proof: 5/6
– Moving a vertex of Kn-k,k from the set of size k
to the set of siz...
Graph Theory
Ch. 1. Fundamental Concept
93
Proof: 6/6
– The product is then n2
/4 for even n and (n2
-1)/4
for odd n. Thus...
Graph Theory
Ch. 1. Fundamental Concept
94
Degree sequence 1.3.27
 The Degree Sequence of a graph is the list of
vertex d...
Graph Theory
Ch. 1. Fundamental Concept
95
Proposition: The nonnegative integers d1 ,…, dn are the
vertex degrees of some ...
Graph Theory
Ch. 1. Fundamental Concept
96
Proof: 2/2 Sufficiency
 Suppose that ∑ di is even.
 We construct a graph with...
Graph Theory
Ch. 1. Fundamental Concept
97
Graphic Sequence 1.3.29
 A graphic sequence is a list of nonnegative
numbers t...
Graph Theory
Ch. 1. Fundamental Concept
98
Recursive condition 1.3.30
 The lists (2, 2, 1, 1) and (1, 0, 1) are graphic. ...
Graph Theory
Ch. 1. Fundamental Concept
99
Recursive condition 1.3.30
 Similarly, to test 33333221, we seek a
realization...
Graph Theory
Ch. 1. Fundamental Concept
100
Recursive condition 1.3.30
This exists if and only if 2223221 is graphic. (See...
Graph Theory
Ch. 1. Fundamental Concept
101
Recursive condition 1.3.30
33333221 3222221 221111
11100
2223221 111221 10111
...
Graph Theory
Ch. 1. Fundamental Concept
102
Theorem. For n>1, an integer list d of size n is graphic if and
only if d’ is ...
Graph Theory
Ch. 1. Fundamental Concept
103
Theorem. For n>1, an integer list d of size n is graphic if and
only if d’ is ...
Graph Theory
Ch. 1. Fundamental Concept
104
Theorem 1.3.31 continue
 To prove necessity, 3/6
– Given a simple graph G rea...
Graph Theory
Ch. 1. Fundamental Concept
105
Theorem 1.3.31
Proof: continue 4/6
– If N(w)=S, then we delete w to obtain G’....
Graph Theory
Ch. 1. Fundamental Concept
106
Theorem 1.3.31
Proof: continue 5/6
– Otherwise,
• Some vertex of S is missing ...
Graph Theory
Ch. 1. Fundamental Concept
Theorem 1.3.31
Proof: continue 6/6
• To find the modification when N(w)≠S , we cho...
Graph Theory
Ch. 1. Fundamental Concept
108
2-switch 1.3.32
 A 2-switch is the replacement of a pair of
edges xy and zw i...
Graph Theory
Ch. 1. Fundamental Concept
109
Theorem: If G and H are two simple graphs with vertex set V,
then dG(v)=dH(v) ...
Graph Theory
Ch. 1. Fundamental Concept
110
Theorem. 1.3.33 (Continue)
 Consider n≥4 , and let w be a vertex of maximum
d...
Graph Theory
Ch. 1. Fundamental Concept
111
Theorem. 1.3.33 Continue
 By the induction hypothesis, some sequence of 2-
sw...
Graph Theory
Ch. 1. Fundamental Concept
112
Directed Graph and Its edges 1.4.2
 A directed graph or digraph G is a triple...
Graph Theory
Ch. 1. Fundamental Concept
113
Directed Graph and its edges 1.4.2
 As with graphs, we
– assign each vertex a...
Graph Theory
Ch. 1. Fundamental Concept
114
Directed Graph and its edges 1.4.2
 When a digraph models a relation, each or...
Graph Theory
Ch. 1. Fundamental Concept
115
Loop and multiple edges in directed graph 1.4.3
 In a graph, a loop is an edg...
Graph Theory
Ch. 1. Fundamental Concept
116
Loop and multiple edges in directed graph 1.4.3
 In the simple digraph, we wr...
Graph Theory
Ch. 1. Fundamental Concept
117
Path and Cycle in Digraph 1.4.6
 A digraph is a path if it is a simple digrap...
Graph Theory
Ch. 1. Fundamental Concept
118
Underlying graph 1.4.9
 The underlying graph of a digraph D:
– the graph G ob...
Graph Theory
Ch. 1. Fundamental Concept
119
Underlying graph 1.4.9
 Most ideals and methods of graph theorem
arise in the...
Graph Theory
Ch. 1. Fundamental Concept
120
Adjacency Matrix and Incidence Matrix
of a Digraph 1.4.10
 In the adjacency m...
Graph Theory
Ch. 1. Fundamental Concept
121
Example of adjacency matrix 1.4.11
 The underlying graph of the digraph below...
Graph Theory
Ch. 1. Fundamental Concept
122
Connected Digraph 1.4.12
 To define connected digraphs, two options
come to m...
Graph Theory
Ch. 1. Fundamental Concept
123
Weakly and strongly connected digraphs 1.4.12
 A graph is weakly connected if...
Graph Theory
Ch. 1. Fundamental Concept
124
Eulerian Digraph 1.4.22
 An Eulerian trail in digraph (or graph) is a
trail c...
Graph Theory
Ch. 1. Fundamental Concept
125
Lemma. If G is a digraph with δ+
(G)≥1, then G
contains a cycle. The same conc...
Graph Theory
Ch. 1. Fundamental Concept
126
Theorem: A digraph is Eulerian if and only if
d+
(v)=d-
(v) for each vertex v ...
Graph Theory
Ch. 1. Fundamental Concept
127
De Bruijn cycles 1.4.25
 Application:
– There are 2n
binary strings of length...
Graph Theory
Ch. 1. Fundamental Concept
128
De Bruijn cycles 1.4.25
 We can use such an arrangement to keep track of the
...
Graph Theory
Ch. 1. Fundamental Concept
129
De Bruijn cycles 1.4.25
 To obtain such a circular arrangement,
– define a di...
Graph Theory
Ch. 1. Fundamental Concept
130
De Bruijn cycles 1.4.25
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
...
Graph Theory
Ch. 1. Fundamental Concept
131
De Bruijn cycles 1.4.25
 We next prove that Dn is Eulerian and show how
an Eu...
Graph Theory
Ch. 1. Fundamental Concept
132
Theorem. The digraph Dn of Application 1.4.25 is Eulerian, and
the edge labels...
Graph Theory
Ch. 1. Fundamental Concept
133
Theorem. The digraph Dn is Eulerian 1.4.26
Proof: 1/2
 Every vertex has out-d...
Graph Theory
Ch. 1. Fundamental Concept
134
Theorem. The digraph Dn is Eulerian 1.4.26
Proof: 2/2
 Also, Dn is strongly c...
Graph Theory
Ch. 1. Fundamental Concept
135
Theorem. The labels on the edges in any Eulerian circuit of Dn
form a cyclic a...
Graph Theory
Ch. 1. Fundamental Concept
136
Theorem. The labels on the edges in any Eulerian circuit of Dn
form a cyclic a...
Graph Theory
Ch. 1. Fundamental Concept
137
Theorem. The labels on the edges in any Eulerian circuit of Dn
form a cyclic a...
Graph Theory
Ch. 1. Fundamental Concept
138
Theorem. The labels on the edges in any Eulerian circuit of
Dn form a cyclic a...
Graph Theory
Ch. 1. Fundamental Concept
139
Theorem. The labels on the edges in any Eulerian circuit of
Dn form a cyclic a...
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Graph theory

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Graph theory

  1. 1. 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
  2. 2. Graph Theory Ch. 1. Fundamental Concept 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
  3. 3. Graph Theory Ch. 1. Fundamental Concept 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
  4. 4. Graph Theory Ch. 1. Fundamental Concept 4 General Model  A vertex : an object  An edge : a relation between two objects common member Committee 1 Committee 2
  5. 5. Graph Theory Ch. 1. Fundamental Concept 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
  6. 6. Graph Theory Ch. 1. Fundamental Concept 6 Loop, Multiple edges  Loop : An edge whose endpoints are equal  Multiple edges : Edges have the same pair of endpoints loop Multiple edges
  7. 7. Graph Theory Ch. 1. Fundamental Concept 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.
  8. 8. Graph Theory Ch. 1. Fundamental Concept 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
  9. 9. Graph Theory Ch. 1. Fundamental Concept 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
  10. 10. Graph Theory Ch. 1. Fundamental Concept 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
  11. 11. Graph Theory Ch. 1. Fundamental Concept 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
  12. 12. Graph Theory Ch. 1. Fundamental Concept 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
  13. 13. Graph Theory Ch. 1. Fundamental Concept 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
  14. 14. Graph Theory Ch. 1. Fundamental Concept 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
  15. 15. Graph Theory Ch. 1. Fundamental Concept 15 Scheduling and graph Coloring 1  Two committees can not hold meetings at the same time if two committees have common member common member Committee 1 Committee 2
  16. 16. Graph Theory Ch. 1. Fundamental Concept 16 Scheduling and graph Coloring 1  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 Committee 1 Committee 2
  17. 17. Graph Theory Ch. 1. Fundamental Concept 17 Scheduling and graph Coloring 2  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
  18. 18. Graph Theory Ch. 1. Fundamental Concept 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
  19. 19. Graph Theory Ch. 1. Fundamental Concept 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.
  20. 20. Graph Theory Ch. 1. Fundamental Concept 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
  21. 21. Graph Theory Ch. 1. Fundamental Concept 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
  22. 22. Graph Theory Ch. 1. Fundamental Concept 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
  23. 23. Graph Theory Ch. 1. Fundamental Concept 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
  24. 24. Graph Theory Ch. 1. Fundamental Concept 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
  25. 25. Graph Theory Ch. 1. Fundamental Concept 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
  26. 26. Graph Theory Ch. 1. Fundamental Concept 26 Complete Graph  Complete Graph : a simple graph whose vertices are pairwise adjacent Complete Graph
  27. 27. Graph Theory Ch. 1. Fundamental Concept 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
  28. 28. Graph Theory Ch. 1. Fundamental Concept 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
  29. 29. Graph Theory Ch. 1. Fundamental Concept 29 Petersen Graph 1.1.37  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
  30. 30. Graph Theory Ch. 1. Fundamental Concept 30 Petersen Graph 1.1.36  Three drawings
  31. 31. Graph Theory Ch. 1. Fundamental Concept 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
  32. 32. Graph Theory Ch. 1. Fundamental Concept 32 Girth 1.1.39, 1.1.40  Girth : the length of its shortest cycle. – If no cycles, girth is infinite
  33. 33. Graph Theory Ch. 1. Fundamental Concept 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.
  34. 34. Graph Theory Ch. 1. Fundamental Concept 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.
  35. 35. Graph Theory Ch. 1. Fundamental Concept 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.
  36. 36. Graph Theory Ch. 1. Fundamental Concept 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
  37. 37. Graph Theory Ch. 1. Fundamental Concept 37 Lemma: Every u,v-walk contains a u,v-path 1.2.5 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.
  38. 38. Graph Theory Ch. 1. Fundamental Concept 38 Lemma: Every u,v-walk contains a u,v-path 1.2.5 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
  39. 39. Graph Theory Ch. 1. Fundamental Concept 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
  40. 40. Graph Theory Ch. 1. Fundamental Concept 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
  41. 41. Graph Theory Ch. 1. Fundamental Concept 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
  42. 42. Graph Theory Ch. 1. Fundamental Concept 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
  43. 43. Graph Theory Ch. 1. Fundamental Concept 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
  44. 44. Graph Theory Ch. 1. Fundamental Concept 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
  45. 45. Graph Theory Ch. 1. Fundamental Concept 45 Induced subgraph 1.2.12  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
  46. 46. Graph Theory Ch. 1. Fundamental Concept 46 Induced subgraph 1.2.12  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
  47. 47. Graph Theory Ch. 1. Fundamental Concept 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.
  48. 48. Graph Theory Ch. 1. Fundamental Concept 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.
  49. 49. Graph Theory Ch. 1. Fundamental Concept 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
  50. 50. Graph Theory Ch. 1. Fundamental Concept 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.
  51. 51. Graph Theory Ch. 1. Fundamental Concept 51 Lemma: Every closed odd walk contains an odd cycle 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
  52. 52. Graph Theory Ch. 1. Fundamental Concept 52 Lemma: Every closed odd walk contains an odd cycle 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
  53. 53. Graph Theory Ch. 1. Fundamental Concept 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
  54. 54. Graph Theory Ch. 1. Fundamental Concept 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
  55. 55. Graph Theory Ch. 1. Fundamental Concept 55 Theorem: A graph is bipartite if it has no odd cycle. 1.2.18 Proof: (sufficiency2/3)  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.
  56. 56. Graph Theory Ch. 1. Fundamental Concept 56 Theorem: A graph is bipartite if it has no odd cycle. 1.2.18 Proof: (sufficiency3/3)  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
  57. 57. Graph Theory Ch. 1. Fundamental Concept 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
  58. 58. Graph Theory Ch. 1. Fundamental Concept 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.
  59. 59. Graph Theory Ch. 1. Fundamental Concept 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].
  60. 60. Graph Theory Ch. 1. Fundamental Concept 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.
  61. 61. Graph Theory Ch. 1. Fundamental Concept 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
  62. 62. Graph Theory Ch. 1. Fundamental Concept 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)
  63. 63. Graph Theory Ch. 1. Fundamental Concept 63 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)  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
  64. 64. Graph Theory Ch. 1. Fundamental Concept 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 →
  65. 65. Graph Theory Ch. 1. Fundamental Concept 65 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 2/3)  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. →
  66. 66. Graph Theory Ch. 1. Fundamental Concept 66 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 3/3)  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.
  67. 67. Graph Theory Ch. 1. Fundamental Concept 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
  68. 68. Graph Theory Ch. 1. Fundamental Concept 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.
  69. 69. Graph Theory Ch. 1. Fundamental Concept 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
  70. 70. Graph Theory Ch. 1. Fundamental Concept 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
  71. 71. Graph Theory Ch. 1. Fundamental Concept 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
  72. 72. Graph Theory Ch. 1. Fundamental Concept 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 }.
  73. 73. Graph Theory Ch. 1. Fundamental Concept 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.
  74. 74. Graph Theory Ch. 1. Fundamental Concept 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
  75. 75. Graph Theory Ch. 1. Fundamental Concept 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
  76. 76. Graph Theory Ch. 1. Fundamental Concept 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
  77. 77. Graph Theory Ch. 1. Fundamental Concept 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
  78. 78. Graph Theory Ch. 1. Fundamental Concept 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.
  79. 79. Graph Theory Ch. 1. Fundamental Concept 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.
  80. 80. Graph Theory Ch. 1. Fundamental Concept 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 )
  81. 81. Graph Theory Ch. 1. Fundamental Concept 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 ∩ = + − ∪ − −≥ + − − =
  82. 82. Graph Theory Ch. 1. Fundamental Concept 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.
  83. 83. Graph Theory Ch. 1. Fundamental Concept 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
  84. 84. Graph Theory Ch. 1. Fundamental Concept 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
  85. 85. Graph Theory Ch. 1. Fundamental Concept 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 .
  86. 86. Graph Theory Ch. 1. Fundamental Concept 86 Example(Cont.) 1.3.20 a b c de f g h switching a would pick up two edges but lose three
  87. 87. Graph Theory Ch. 1. Fundamental Concept 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
  88. 88. Graph Theory Ch. 1. Fundamental Concept 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
  89. 89. Graph Theory Ch. 1. Fundamental Concept 89 Theorem: The maximum number of edges in an n- vertex triangle free simple graph is  n2 /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
  90. 90. Graph Theory Ch. 1. Fundamental Concept 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 Theorem: The maximum number of edges in an n- vertex triangle free simple graph is  n2 /4  1.3.23 Proof: 3/6
  91. 91. Graph Theory Ch. 1. Fundamental Concept 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| Theorem: The maximum number of edges in an n- vertex triangle free simple graph is  n2 /4  1.3.23 n-k k
  92. 92. Graph Theory Ch. 1. Fundamental Concept 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 . Theorem: The maximum number of edges in an n- vertex triangle free simple graph is  n2 /4  1.3.23 n-k k
  93. 93. Graph Theory Ch. 1. Fundamental Concept 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. Theorem: The maximum number of edges in an n- vertex triangle free simple graph is  n2 /4  1.3.23
  94. 94. Graph Theory Ch. 1. Fundamental Concept 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)
  95. 95. Graph Theory Ch. 1. Fundamental Concept 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.
  96. 96. Graph Theory Ch. 1. Fundamental Concept 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
  97. 97. Graph Theory Ch. 1. Fundamental Concept 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.
  98. 98. Graph Theory Ch. 1. Fundamental Concept 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
  99. 99. Graph Theory Ch. 1. Fundamental Concept 99 Recursive condition 1.3.30  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
  100. 100. Graph Theory Ch. 1. Fundamental Concept 100 Recursive condition 1.3.30 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.
  101. 101. Graph Theory Ch. 1. Fundamental Concept 101 Recursive condition 1.3.30 33333221 3222221 221111 11100 2223221 111221 10111 wv u u v u
  102. 102. Graph Theory Ch. 1. Fundamental Concept 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’
  103. 103. Graph Theory Ch. 1. Fundamental Concept 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
  104. 104. Graph Theory Ch. 1. Fundamental Concept 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
  105. 105. Graph Theory Ch. 1. Fundamental Concept 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
  106. 106. Graph Theory Ch. 1. Fundamental Concept 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
  107. 107. Graph Theory Ch. 1. Fundamental Concept 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
  108. 108. Graph Theory Ch. 1. Fundamental Concept 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
  109. 109. Graph Theory Ch. 1. Fundamental Concept 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.
  110. 110. Graph Theory Ch. 1. Fundamental Concept 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
  111. 111. Graph Theory Ch. 1. Fundamental Concept 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*.
  112. 112. Graph Theory Ch. 1. Fundamental Concept 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.
  113. 113. Graph Theory Ch. 1. Fundamental Concept 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.
  114. 114. Graph Theory Ch. 1. Fundamental Concept 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.
  115. 115. Graph Theory Ch. 1. Fundamental Concept 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
  116. 116. Graph Theory Ch. 1. Fundamental Concept 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
  117. 117. Graph Theory Ch. 1. Fundamental Concept 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.
  118. 118. Graph Theory Ch. 1. Fundamental Concept 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
  119. 119. Graph Theory Ch. 1. Fundamental Concept 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.
  120. 120. Graph Theory Ch. 1. Fundamental Concept 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.
  121. 121. Graph Theory Ch. 1. Fundamental Concept 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
  122. 122. Graph Theory Ch. 1. Fundamental Concept 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.
  123. 123. Graph Theory Ch. 1. Fundamental Concept 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.
  124. 124. Graph Theory Ch. 1. Fundamental Concept 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.
  125. 125. Graph Theory Ch. 1. Fundamental Concept 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.
  126. 126. Graph Theory Ch. 1. Fundamental Concept 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
  127. 127. Graph Theory Ch. 1. Fundamental Concept 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
  128. 128. Graph Theory Ch. 1. Fundamental Concept 128 De Bruijn cycles 1.4.25  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.
  129. 129. Graph Theory Ch. 1. Fundamental Concept 129 De Bruijn cycles 1.4.25  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.
  130. 130. Graph Theory Ch. 1. Fundamental Concept 130 De Bruijn cycles 1.4.25 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..
  131. 131. Graph Theory Ch. 1. Fundamental Concept 131 De Bruijn cycles 1.4.25  We next prove that Dn is Eulerian and show how an Eulerian circuit yields the desired circular arrangement.
  132. 132. Graph Theory Ch. 1. Fundamental Concept 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.
  133. 133. Graph Theory Ch. 1. Fundamental Concept 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
  134. 134. Graph Theory Ch. 1. Fundamental Concept 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
  135. 135. Graph Theory Ch. 1. Fundamental Concept 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
  136. 136. Graph Theory Ch. 1. Fundamental Concept 136 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: 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
  137. 137. Graph Theory Ch. 1. Fundamental Concept 137 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: 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
  138. 138. Graph Theory Ch. 1. Fundamental Concept 138 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: 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
  139. 139. Graph Theory Ch. 1. Fundamental Concept 139 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: 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.

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