Chapter 04 - Graph theory bài toán graph trong toán rời rạc

Discrete Mathematics
Graph
Hà Minh Hoàng
Faculty of Data Science and Artificial Intelligence
College of Technology
National Economics University

● Graphs and Their Classifications
● Graph Terminologies
● Graph Representations and Isomorphism
● Paths and Connectivity
● Eulerian and Hamiltonian Paths
● Shortest Path Problem
● Planar Graphs
● Graph Coloring
Content

- Graph theory is a well-established field of science with numerous modern
applications.
- Graphs are used to solve problems in various domains such as electrical
circuits, chemical compound structures, computer networks, and more.
- A graph is a discrete structure consisting of vertices and edges connecting
those vertices.
- Graphs are classified based on the properties of the edges connecting pairs
of vertices.
Graphs and Their Classifications

- A simple graph =( , ) consists of a non-empty set whose elements are
𝐺 𝑉 𝐸 𝑉
called vertices, and a set whose elements are called edges. Each edge is
𝐸
an unordered pair of distinct vertices.
Classification: simple graph

- A multigraph =( , ), where is the set of vertices and is the set of
𝐺 𝑉 𝐸 𝑉 𝐸
edges, is an undirected graph that may have multiple edges connecting the
same pair of vertices (called parallel edges or multiple edges).
- A simple graph is a special case of a multigraph.
Classification: multigraph

Classification: simple graph vs multigraph

- A pseudograph =( , ), where is the set of vertices and is the set of
𝐺 𝑉 𝐸 𝑉 𝐸
edges, is a graph that allows edges connecting a vertex to itself (called
loops).
- A pseudograph can have multiple edges and loops.
Classification: pseudograph

- A simple directed graph =( ,
𝐺 𝑉 A), where is a non-empty set whose
𝑉
elements are called vertices, and A is a set whose elements are called arcs,
consists of ordered pairs of distinct vertices.
Classification: simple directed graph

- A directed multigraph =( ,
𝐺 𝑉 A), where is the set of vertices and
𝑉 A is the
set of arcs, is a graph consisting of directed arcs, and multiple directed arcs
between the same pair of vertices (called parallel arcs) are allowed.
- A simple directed graph is a special case of a directed multigraph.
Classification: directed multigraph

Example 1. Niche overlap graph in ecology.
- Graphs are used in many models that consider the interactions between species,
e.g., the competition among species in an ecosystem can be modeled using a "niche"
graph.
- Each species is represented by a vertex. An undirected edge connects two vertices
if the corresponding species compete with each other (i.e., they share the same food
source).
- From this graph, squirrels and raccoons compete with each other, while crows and
shrews do not.
Examples: Biological networks

Example 2. Protein interaction graph.
- A protein interaction in a living cell occurs when two or more proteins in that cell bind to
perform a biological function.
- Protein interactions are crucial for most biological functions, many scientists work on
discovering new proteins and understanding interactions between proteins- Protein
interactions within a cell can be modeled using a protein interaction graph (also called a
protein–protein interaction network)
- An undirected graph in which each protein is represented by a vertex, with an edge
connecting the vertices representing each pair of proteins that interact.
- Protein interaction graphs can be used to deduce important biological information, such as
by identifying the most important proteins for various functions and the functionality of
newly discovered proteins.

Example 2. Protein interaction graph.
- The protein interaction graph of a cell is extremely large and complex.
- Yeast cells have more than 6,000 proteins, and more than 80,000 interactions
- Human cells have more than 100,000 proteins, with perhaps 1,000,000 interactions between them.
- Additional vertices and edges are added to a protein interaction graph when new
proteins and interactions between proteins are discovered.
- The protein interaction graphs are often split into smaller graphs called modules
that represent groups of proteins that are involved in a particular function of a cell.
- Figure illustrates a module of the protein interaction graph, comprising the
complex of proteins that degrade RNA in human cells.

Example 2. Influence Graph
- When studying the personalities of a group of people, it is observed that some
individuals may influence the thoughts of others.
- A directed graph, known as an influence graph, can be used to model this problem.
- Each person in the group is represented by a vertex.
- If a person represented by vertex can influence a person represented by vertex
𝑎
, then there is a directed arc from vertex to vertex .
𝑏 𝑎 𝑏
Examples: Social networks

Other social networks:
- Acquaintanceship and Friendship Graphs.
- Collaboration Graphs: Academic collaboration graph, Hollywood graph, in sports.
Examples: Social networks

Call graphs:
Examples: Communication networks

Example 3. Citation graph
- A citation graph is a directed graph where each node represents a scientific paper,
and each directed edge points from one paper to another that it cites.
- These graphs form the backbone of scholarly communication and offer powerful
insights into the structure and evolution of scientific knowledge.
- The Web graph
Examples: Information graphs

- The Seven Bridges of Königsberg is a famous historical problem in mathematics and is
considered the foundation of graph theory.
- The city of Königsberg (now Kaliningrad, Russia) was set on both sides of the Pregel River
and included two large islands connected to each other and the mainland by seven bridges.
- The challenge posed to citizens was:"Is it possible to walk through the city in such a way that
each bridge is crossed exactly once?“
- In 1736, Leonhard Euler proved that such a walk was not possible.
Examples: Transportation networks

- Airline routes
- Road networks
- Rail networks
- Shipping networks
Examples: Transportation networks

- Module dependency graphs
- Precedence graph and concurrent processing
Examples: Software design applications

- Round-robin tournaments
- Single-elimination tournaments
Examples: Tournaments

- Exercise 3: Construct an influence graph for the board members of a company if
the President can influence the Director of Research and Development, the
Director of Marketing, and the Director of Operations; the Director of Research and
Development can influence the Director of Operations; the Director of Marketing
can influence the Director of Operations; and no one can influence, or be
influenced by, the Chief Financial Officer.
Exercises

- Exercise 4: Explain how the two telephone call graphs for calls made during the
month of January and calls made during the month of February can be used to
determine the new telephone numbers of people who have changed their
telephone numbers.
Exercises

- Definition 1:
Two vertices u and v in an undirected graph G are called adjacent (or neighbors) in G if u and
v are endpoints of an edge e of G. Such an edge e is called incident with the vertices u and v
and e is said to connect u and v.
- Definition 2:
The set of all neighbors of a vertex v of G = (V ,E), denoted by N(v), is called the
neighborhood of v. If A is a subset of V, we denote by N(A) the set of all vertices in G that are
adjacent to at least one vertex in A.
- Definition 3:
The degree of a vertex in an undirected graph is the number of edges incident with it, except
that a loop at a vertex contributes twice to the degree of that vertex. The degree of the vertex v
is denoted by deg(v).
Graph Terminologies

What are the degrees and what are the neighborhoods of the vertices in the graphs G
and H displayed in Figure.
Graph Terminologies

Theorem 1:
Graph Terminologies

How many edges are there in a simple graph with 10 vertices, each of degree 5?
How many edges are there in a simple graph with 99 vertices, each of degree 5?
Graph Terminologies

Theorem 2: An undirected graph has an even number of vertices of odd degree.
Graph Terminologies

Definition 4: When (u, v) is an edge of the graph G with directed edges, u is said to be
adjacent to v and v is said to be adjacent from u. The vertex u is called the initial
vertex of (u, v), and v is called the terminal or end vertex of (u, v). The initial vertex
and terminal vertex of a loop are the same.
Definition 5: In a graph with directed edges the in-degree of a vertex v, denoted by
deg−(v), is the number of edges with v as their terminal vertex. The out-degree of v,
denoted by deg+(v), is the number of edges with v as their initial vertex. (Note that a
loop at a vertex contributes 1 to both the in-degree and the out-degree of this vertex.)
Graph Terminologies

Find the in-degree and out-degree of each vertex in the graph G with directed edges
shown in Figure
Graph Terminologies

Theorem 3:
Graph Terminologies

Complete Graphs
A complete graph on n vertices, denoted by Kn, is a simple graph that contains exactly
one edge between each pair of distinct vertices. The graphs Kn, for n = 1, 2, 3, 4, 5, 6,
are displayed in Figure.
Graph Terminologies

Cycles
A cycle Cn, n ≥ 3, consists of n vertices v1, v2, . . . , vn and edges {v1, v2}, {v2, v3}, . . . , {vn−1,
vn}, and {vn, v1}. The cycles C3, C4, C5, and C6 are displayed in Figure
Graph Terminologies

Wheels
We obtain a wheel Wn when we add an additional vertex to a cycle Cn, for n ≥ 3, and
connect this new vertex to each of the n vertices in Cn, by new edges. The wheels W3, W4,
W5, and W6 are displayed in Figure.
Graph Terminologies

n-Cubes
An n-dimensional hypercube, or n-cube, denoted by Qn, is a graph that has vertices
representing the 2n bit strings of length n. Two vertices are adjacent if and only if the bit
strings that they represent differ in exactly one bit position. We display Q1, Q2, and Q3 in
Figure.
Graph Terminologies

A simple graph G is called bipartite if its vertex set V can be partitioned into two disjoint
sets V1 and V2 such that every edge in the graph connects a vertex in V1 and a vertex in V2
(so that no edge in G connects either two vertices in V1 or two vertices in V2). When this
condition holds, we call the pair (V1, V2) a bipartition of the vertex set V of G.
Example: C6 is a bipartite graph.
Graph Terminologies

Example: Is K3 is a bipartite graph?
Graph Terminologies

Theorem: A simple graph is bipartite if and only if it is possible to assign one of two
different colors to each vertex of the graph so that no two adjacent vertices are assigned
the same color.
Exercise:
Graph Terminologies

Complete Bipartite Graphs
A complete bipartite graph Km,n is a graph that has its vertex set partitioned into two
subsets of m and n vertices, respectively with an edge between two vertices if and only if
one vertex is in the first subset and the other vertex is in the second subset. The complete
bipartite graphs K2,3, K3,3, K3,5, and K2,6 are displayed in Figure
Graph Terminologies

Bipartite graphs and matchings
Job assignment
Graph Terminologies

Bipartite graphs and matchings
Marriages on an Island
● Suppose that there are m men and n women on an island. Each person has a
list of members of the opposite gender acceptable as a spouse.
● Construct a bipartite graph G = (V1, V2) where V1 is the set of men and V2 is the
set of women so that there is an edge between a man and a woman if they
find each other acceptable as a spouse.
● A matching in this graph consists of a set of edges, where each pair of
endpoints of an edge is a husband-wife pair.
● A maximum matching is a largest possible set of married couples, and a
complete matching of V1 is a set of married couples where every man is
married, but possibly not all women.
Graph Terminologies

Local area networks
Star topology, ring topology and hybrid topology
Some Applications of Special Types of Graphs

Definition 7:
Graph Terminologies

Definition 8: Let G = (V ,E) be a simple graph. The subgraph induced by a subsetW of the
vertex set V is the graph (W, F), where the edge set F contains an edge in E if and only if
both endpoints of this edge are in W.
- The graph G shown in Figure is a subgraph of K5. If we add the edge connecting c and e
to G, we obtain the subgraph induced by W = {a, b, c, e}.
Graph Terminologies

Definition 9: The union of two simple graphs G1 = (V1, E1) and G2 = (V2, E2) is the simple
graph with vertex set V1 V
∪ 2 and edge set E1 E
∪ 2. The union of G1 and G2 is denoted by
G1 G
∪ 2.
Graph Terminologies

For which values of n are these graphs bipartite?
a) Kn
b) Cn
c) Wn
d) Qn
Exercises

- List all the edges of the graph (without multiple edges)
- Use adjacency lists which specify the vertices that are adjacent to each vertex of the
graph (without multiple edges)
Representing Graphs
All the edges
1. {a,b}
2. {a,c}
3. {a,e}
4. {c,d}
5. {c,e}
6. {e,d}

- Adjacency matrices (with or without multiple edges)
Representing Graphs

- Adjacency matrices (with multiple edges)
Representing Graphs

Adjacency matrices vs adjacency list
- Sparse graph: adjacency lists are preferable
- If each vertex has degree not exceeding c, no more than cn items in adjacency list, where n is the number of
vertices in the graph
- In the adjacency matrices, there are n2
items
- Sparse matrices
- Dense graph: adencecy matrices are preferable
- Check if there is an edge (vi, vj) in the graph
Representing Graphs

- How about representing directed graphs?
Representing Graphs

- Incidence matrices
- n × m matrix M = [mij ]
Representing Graphs

- Incidence matrices
- Multiple edges and loops?
Representing Graphs

- The simple graphs G1 = (V1, E1) and G2 = (V2, E2) are isomorphic if there exists a one
to-one and onto function f from V1 to V2 with the property that a and b are adjacent in
G1 if and only if f(a) and f(b) are adjacent in G2, for all a and b in V1. Such a function f is
called an isomorphism.
- Two simple graphs that are not isomorphic are called nonisomorphic.
- In other words, when two simple graphs are isomorphic, there is a one-to-one
correspondence between vertices of the two graphs that preserves the adjacency
relationship.
Isomorphism of Graphs

- Show that the graphs G = (V ,E) and H =
(W, F) are isomorphic.

- Show that the graphs G = (V ,E) and H = (W, F)
are isomorphic.
- The function f with f(u1) = v1, f(u2) = v4, f(u3) = v3,
and f(u4) = v2 is a one-to-one correspondence
between V and W

Determining whether Two Simple Graphs are Isomorphic.
- It is often difficult to determine whether two simple graphs are isomorphic.
- n! possible one-to-one correspondences between the vertex sets of two simple graphs with n vertices.
- Sometimes it is not hard to show that two graphs are not isomorphic.
- We can show that two graphs are not isomorphic if we can find a property only one of the two graphs
has, but that is preserved by isomorphism.
- A property preserved by isomorphism of graphs is called a graph invariant.
- Isomorphic simple graphs must have the same number of vertices
- Isomorphic simple graphs also must have the same number of edges,
- The degrees of the vertices in isomorphic simple graphs must be the same.

Exercises:

Paths (walk) and circuits (closed walk):
- Let n be a nonnegative integer and G an undirected graph. A path of length n from u to
v in G is a sequence of n edges e1, . . . , en of G for which there exists a sequence x0 =
u, x1, . . . , xn−1, xn = v of vertices such that ei has, for i = 1, . . . , n, the endpoints xi−1
and xi. When the graph is simple, we denote this path by its vertex sequence x0, x1, . . . ,
xn (because listing these vertices uniquely determines the path).
- The path is a circuit if it begins and ends at the same vertex, that is, if u = v, and has
length greater than zero.
- A path or circuit is simple if it does not contain the same edge more than once.
Connectivity

Paths represent useful information in many graph models
- Acquaintanceship graphs:
- Six Degrees of Separation: a famous social theory suggests that any two people in the world can be
connected through at most six intermediate relationships.
- First introduced by Hungarian author Frigyes Karinthy in 1929 in his short story "Chains".
- Gained popularity through experiments by Stanley Milgram (1960s), who had participants try to send a
letter to a target person via acquaintances — the average number of intermediaries was 5–6.
- The play by John Guare helped bring the idea of “Six Degrees of Separation” into mainstream culture,
beyond academic or psychological discussions.
- Studies on Facebook show that only about 3.5 to 4 intermediaries are needed to connect any two
users.
Connectivity

Paths represent useful information in
many graph models: Collaboration graphs
– In Academic: Erdos number
- Càng gần Erdős, bạn càng được
coi là gần “trung tâm toán học thế
giới”
- Những người có Erdos number ≤ 4
thường rất được ngưỡng mộ trong
cộng đồng nghiên cứu toán.
- In Hollywood graph: Bacon number
Connectivity

- An undirected graph is called connected if there is a path between every pair of
distinct vertices of the graph.
- An undirected graph that is not connected is called disconnected.
- We say that we disconnect a graph when we remove vertices or edges, or both, to
produce a disconnected subgraph.
- Any two computers in the network can communicate if and only if the graph of this
network is connected.
Connectedness in Undirected Graphs

- An undirected graph is called connected if there is a
path between every pair of distinct vertices of the graph.
- An undirected graph that is not connected is called
disconnected.
- We say that we disconnect a graph when we remove
vertices or edges, or both, to produce a disconnected
subgraph.
- Any two computers in the network can communicate if
and only if the graph of this network is connected.

- Theorem: There is a simple path between every pair of distinct vertices of a
connected undirected graph.
- Proof ???

- A connected component of a graph G is a connected subgraph of G that is not
a proper subgraph of another connected subgraph of G. That is, a connected
component of a graph G is a maximal connected subgraph of G. A graph G that
is not connected has two or more connected components that are disjoint and
have G as their union.

How connected is a graph?
- The removal from a graph of a vertex and all incident edges produces a
subgraph with more connected components. Such vertices are called cut
vertices (or articulation points).
- An edge whose removal produces a graph with more connected components
than in the original graph is called a cut edge or bridge.
- In a graph representing a computer network, a cut vertex and a cut edge
represent an essential router and an essential link that cannot fail for all
computers to be able to communicate.

Cut vertex and cut edge
- Find the cut vertices and cut edges in G1 and G2

Vertex connectivity
- Not all graphs have cut vertices. For example, Kn, where n ≥ 3
- Connected graphs without cut vertices are called nonseparable graphs, and can
be thought of as more connected than those with a cut vertex.
- A measure of graph connectivity based on the minimum number of vertices that
can be removed to disconnect a graph.

Vertex connectivity
- A subset V’ of the vertex set V of G = (V ,E) is a vertex cut, or separating set, if
G − V’ is disconnected.
- For instance, in the graph in Figure, the set {b, c, e} is a vertex cut with three
vertices
- Every connected graph, except a complete graph, has a vertex cut.
- The vertex connectivity of a noncomplete graph G, denoted by κ(G), as the
minimum number of vertices in a vertex cut.

Vertex connectivity
- For every graph G, κ(G) is minimum
number of vertices that can be
removed from G to either disconnect G
or produce a graph with a single vertex
- Compute κ(Kn)?
- The larger κ(G) is, the more connected
we consider G to be.
- A graph is k-connected (or k-vertex-
connected), if κ(G) ≥ k.
- Find the vertex connectivity for each of
the graphs:

Vertex connectivity
- Find the vertex connectivity for each of
the graphs:
- κ(G1) = 1, κ(G2) = 1
- κ(G3) = 2, κ(G4) = 2
- κ(G5) = 3

Edge connectivity
- We can also measure the connectivity of a connected graph G = (V ,E) in terms of
the minimum number of edges that we can remove to disconnect it.
- If a graph has a cut edge, we need only remove it to disconnect G. If G does not
have a cut edge, we look for the smallest set of edges that can be removed to
disconnect it.
- A set of edges E’ is called an edge cut of G if the subgraph G − E’ is disconnected
- The edge connectivity of a graph G, denoted by λ(G), is the minimum number of
edges in an edge cut of G.
- λ(G) = n − 1 where G is a graph with n vertices if and only if G = Kn
- λ(G) ≤ n − 2 when G is not a complete graph.

Edge connectivity
- Find the edge connectivity for each of
the graphs:
- λ(G1) = 1,
- λ(G2) = 2, λ(G3) = 2
- κ(G4) = 3, κ(G5) = 3

- A directed graph is strongly connected if there is a path from a to b and from b
to a whenever a and b are vertices in the graph.
- A directed graph is weakly connected if there is a path between every two
vertices in the underlying undirected graph.
Connectedness in Directed Graphs

Strong components of a directed graph
- The subgraphs of a directed graph G that are strongly connected but not
contained in larger strongly connected subgraphs, that is, the maximal strongly
connected subgraphs, are called the strongly connected components or
strong components of G.
Connectedness in Directed Graphs

- Paths and circuits can help determine whether two graphs are isomorphic.
- The existence of a simple circuit of a particular length is a useful invariant that
can be used to show that two graphs are not isomorphic.
- Paths can be used to construct mappings that may be isomorphisms.
Paths and Isomorphism

Theorem: Let G be a graph with adjacency matrix A with respect to the ordering v1, v2,
. . . , vn of the vertices of the graph (with directed or undirected edges, with multiple
edges and loops allowed). The number of different paths of length r from vi to vj, where
r is a positive integer, equals the (i, j )th entry of Ar
How many paths of length three and four are there from a to d in the simple graph G?
Counting Paths Between Vertices

How many vertices and how many edges do the following graphs have?
a. 𝐾9
b. 𝐶15
c. 𝑊15
d. 𝐾3,4
e. 𝑄7
Exercises

Which of the following graphs is a bipartite graph?
Exercises

Construct and calculate the number of edges of a simple graph with the following
vertex degrees:
• 4, 3, 3, 2, 2
• 3, 3, 3, 3, 2
• 1, 2, 3, 4, 5
• 1, 2, 3, 4, 4
Exercises

Let G be a graph with v vertices and e edges. Let M be the maximum degree of the
vertices of G, and let m be the minimum degree of the vertices of G.
Show that:
a) 2e/v ≥ m.
b) 2e/v ≤ M.
Exercises

a) Find the adjacency matrix of 𝐾2,3
b) Find the number of paths of length 3 and 4 from a degree-3 vertex to a degree-2
vertex.
Exercises

An Euler circuit in a graph G is a simple circuit containing every edge of G. An Euler
path in G is a simple path containing every edge of G.
Euler and Hamilton Paths

Which of the undirected graphs have an Euler circuit? Of those that do not, which
have an Euler path?

A connected multigraph with at least two vertices has an Euler circuit if and only if
each of its vertices has even degree.

A connected multigraph has an Euler path but not an Euler circuit if and only if it has exactly
two vertices of odd degree.
Necessary and sufficient conditions for Euler paths and circuits in directed graphs are given
in Exercises 16 and 17.

- A simple path in a graph G that passes through every vertex exactly once is called a
Hamilton path, and a simple circuit in a graph G that passes through every vertex
exactly once is called a Hamilton circuit.
- The simple path x0, x1, . . . , xn−1, xn in the graph G = (V ,E) is a Hamilton path if V =
{x0, x1, . . . , xn−1, xn} and xi ≠ xj for 0 ≤ i < j ≤ n
- The simple circuit x0, x1, . . . , xn−1, xn, x0 (with n > 0) is a Hamilton circuit if x0,
x1, . . . , xn−1, xn is a Hamilton path.

- A simple path in a graph G that passes through every vertex exactly once is called a
Hamilton path, and a simple circuit in a graph G that passes through every vertex
exactly once is called a Hamilton circuit.
- The simple path x0, x1, . . . , xn−1, xn in the graph G = (V ,E) is a Hamilton path if V =
{x0, x1, . . . , xn−1, xn} and xi ≠ xj for 0 ≤ i < j ≤ n
- The simple circuit x0, x1, . . . , xn−1, xn, x0 (with n > 0) is a Hamilton circuit if x0,
x1, . . . , xn−1, xn is a Hamilton path.
- Hamilton’s “A Voyage Round the World” Puzzle.

- Which of the simple graphs in Figures have a Hamilton circuit or, if not, a Hamilton
path?

- Determine whether the picture shown can be drawn with a pencil in a continuous
motion without lifting the pencil or retracing part of the picture!

- For which values of n do these graphs have an Euler circuit?
a) Kn
b) Cn
c) Wn
d) Qn
- For which values of n do the graphs above have an Euler path but no Euler circuit?

- For which values of m and n does the complete bipartite graph Km,n have an
a) Euler circuit?
b) Euler path?

- Graphs that have a number assigned to each edge are called weighted graphs.
- Weighted graphs are used to model computer networks. Communications costs
(such as the monthly cost of leasing a telephone line), the response times of the
computers over these lines, or the distance between computers, can all be studied
using weighted graphs.
Shortest-Path Problems

- Graphs that have a number assigned to each edge are called weighted graphs.
- Weighted graphs are used to model computer networks. Communications costs
(such as the monthly cost of leasing a telephone line), the response times of the
computers over these lines, or the distance between computers, can all be studied
using weighted graphs.
A Shortest-Path Algorithm

A Shortest-Path Algorithm
- Find the length of a shortest path between these pairs of vertices in the weighted
graph in Exercise 3.
a) a and d
b) a and f
c) c and f
d) b and z

Planar Graphs
- Can a graph be drawn in the plane without edges crossing?
???

Planar Graphs
- A graph is called planar if it can be drawn in the plane without any edges crossing
(where a crossing of edges is the intersection of the lines or arcs representing
them at a point other than their common endpoint)
- Such a drawing is called a planar representation of the graph
- Is K3,3 planar?

Planar Graphs
Applications:
- Design of electronic circuits
- Road networks

Planar Graphs
Euler’s formula:
Let G be a connected planar simple graph with e edges and v vertices. Let r be the number of
regions in a planar representation of G.
Then r = e − v + 2.
- Exercise: Suppose that a connected planar simple graph has 20 vertices, each of degree
3. Into how many regions does a representation of this planar graph split the plane?

Planar Graphs
Prove that:
If G is a connected planar simple graph with e edges and v vertices, where v ≥ 3, then
e ≤ 3v − 6.

Planar Graphs
Prove that:
- If G is a connected planar simple graph, then G has a vertex of degree not exceeding
five.
- K5 is nonplanar?
- If a connected planar simple graph has e edges and v vertices with v ≥ 3 and no
circuits of length three, then e ≤ 2v − 4.
- K3,3 is nonplanar.

Planar Graphs
- K5 , K3,3 are nonplanar. A graph is not planar if it contains either of these two graphs
as a subgraph.
- A planar graph G and every graph obtained from G by removing an edge (u, v) and
adding a vertex w with two edges (u, w) and (w, v) is also planar and is called a
homeomorphic graph of G.
- A graph is nonplanar if and only if it contains a subgraph homeomorphic to K3,3 or K5.

Planar Graphs
- Is the Petersen graph planar?

Graph coloring
Map Coloring
- Neighboring regions are colored
with different colors.
- Use the minimum number of colors.
Represent the map as a graph

Graph coloring
- Vietnam map coloring

Graph coloring
- Definition 1. Coloring a simple graph is the assignment of colors to
its vertices such that no two adjacent vertices are assigned the same
color.
- Definition 2. The chromatic number of a graph is the minimum
number of colors needed to color the graph.
- Theorem 1. Four Color Theorem. The chromatic number of any
planar graph is at most four.

Graph coloring
- Exercise: the number of colors of Kn, Km,n, Cn?

Graph coloring
- The graph coloring problem has many applications
- Application in exam scheduling: Scheduling exams so that no student has to
take two exams at the same time.
- Courses are represented as vertices of the graph.
- Two courses that a student has to take together an edge.
→
- Exam times are represented by different colors.
Scheduling is equivalent to coloring the graph.
→

Chapter 04 - Graph theory bài toán graph trong toán rời rạc

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Chapter 04 - Graph theory bài toán graph trong toán rời rạc