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Discrete and Combinatorial Mathematics

R. P. Grimaldi, 5th edition, 2004

Chapter 7Graph Theory

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In the beginning…Leonhard Euler

Swiss mathematician, 1707-1783. First paper in graph theory. In 1736, he wrote a solution to the

problem of Königsberg bridges.

Problem of the Königsberg bridges: In East Prussia Starting and ending at the same region, is it possible to cross

all seven bridges just once and return to the starting region?

Pregel river

Kneiphof island

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Graphs A graph G = (V, E) consists

V = V(G) : set of vertices (vertex set) E = E(G) VV : set of edges (edge

set)

If E(G) is a set of ordered pairs, then G is called a directed graph; otherwise an undirected graph.

Example: V = {x, u, v, w} E = {(x,v), (x,u), (u,v), (v,w), (u,w)} x

v

u

w

x

v

u

w

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Subgraphs If G = (V, E) is a graph, then G1 = (V1,E1) is

called a subgraph of G if V1 V and E1 E, where each edge in E1 is incident with vertices in V1.

x

v

u

w

x

v

uG G1

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EdgesAn edge is labeled by a pair of vertices, for

instance e = (v,w). e is said to be incident with v and w. v and w are said adjacent.

isolated vertex: a vertex without incident edges. loop: an edge whose endpoints are equal. parallel edges: edges with the same endpoints.

v wisolatedvertex

loop

e

paralleledges

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Representations of graphsAdjacency matrix

Rows and columns are labeled with ordered vertices. Aij = 1, if there is an edge between the row vertex and

the column vertex. Aij = 0 if no edge exists between them.

Incidence matrix Label rows with vertices Label columns with edges Iij = 1 if an edge is incident to a vertex, Iij = 0

otherwise.

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Different graphs

Simple graphA graph without loops or parallel edges.

Weighted graphA graph where each edge is assigned a

numerical label or “weight”.

1

2 4

8

6

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WalksG = (V,E) : an undirected graph,

x ,y V (not necessarily distinct)

x-y walk in G: a loop-free finite alternating sequence

of vertices and edges, where ei = (xi-1,xi). length of the walk = n. x-y walk with x = y: closed walk. Note: A walk may repeat both vertices and edges.

yxexeexexexx nnnn ,,,,,,,,,, 11322110

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Trails, Circuits, and Paths If no edge in the x-y walk is repeated, then the

walk is called an x-y trail. A closed x-y trail is called a circuit. If no vertex of the x-y walk repeated, then the

walk is called an x-y path. A closed x-y path is called a cycle.

x

v

u

wyeveweuevextrail ,,,,,,,,,,: 62531e1

e2

e5

e3

e4

xeueveweyevexcircuit ,,,,,,,,,,,,: 432761

e6 e7

y

yewevexpath ,,,,,,: 721

xeuewevexcycle ,,,,,,,,: 4521

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Connected graphs Let G = (V,E) be an undirected graph. We call G

connected if there is a path between any two distinct vertices of G; otherwise it is called disconnected.

Each connected subgraph of a disconnected graph G is called a component of G.

G H

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Spanning subgraphs Given a grapg G = (V, E), let G1 = (V1,E1) be a

subgraph of G. iIf V1 = V then G1 is called a spanning subgraph of G.

x

v

u

w

x

v

uG G1

w

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Induced subgraphs Given a grapg G = (V, E). If U V, the

subgraph induced by U, denoted by <U>, is the subgraph whose vertex set is U and which contains all edges with both endpoints in U.

x

v

u

w

x

v

uG <U>

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Complete graphs The complete graph Kn is the simple graph

with n vertices and every pair of vertices is joined by an edge.

Example: K5

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Complement graphs Let G be a simple graph with n vertices. The

complement of G, denoted by , is the subgraph of Kn consisting of the n vertices in G and all edges that are not in G.

Example:

G

G G

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Bipartite graphsA simple undirected graph is called bipartite if

there exists a partition of the vertex set so that V1V2= and no edges incident with two vertices in the same subset Vk, k = 1,2.

V1

V2

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Complete bipartite graphs

A bipartite graph G = (V,E) with vertex partition V1 and V2 , where |V1| = m and |V2| = n, is the complete bipartite graph, denoted by Km,n, if every vertex in V1 is joined to a vertex in V2 and vice versa.

V1

V2

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Isomorphic graphs

G1 and G2 are isomorphic if there exist one-to-one onto functions f: V(G1) → V(G2) and g: E(G1) → E(G2) such that an edge e is adjacent to vertices v, w in G1 if and only if g(e) is adjacent to f(v) and f(w) in G2

aq, bv, cu, dy, er, fw, gx, ht, iz, js

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Degree of a vertexThe degree of a vertex v, denoted by deg(v),

is the number of edges incident on v.

Example: deg(a) = 4, deg(b) = 3,

deg(c) = 2, deg(d) = 3.a b

c d

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Sum of vertex degreesTheorem 11.2 If G = (V,E) is an undirected graph,

then

Proof. Each edge contributes exactly two counts, one count (degree) for each endpoint.

#

Corollary 11.1 For any undirected graph, the number of vertices of odd degree must be even.

Vv

Ev .||2)deg(

Vv

Ev .||2)deg(

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Regular graphsAn undirected graph where each vertex has the

same degree is called a regular graph. If deg(v) = k for all vertices v, then the graph is called k-regular.

Hypercube Qn: n regular.010

000

001

100

111

011

110

101

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Euler circuit

Let G = (V,E) be an undirected graph with no isolated vertices. Then G is said to have an Euler circuit if there is a circuit in G that traverses every edge of the graph exactly once.

The Königsberg bridge problem: Starting and ending at the same region, is it

possible to cross all seven bridges just once and return to the starting region?

This problem can be represented by a graph. Edges represent bridges and each vertex represents a region.

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Euler graphsTheorems 11.3 Let G = (V,E) be an undirected

graph with no isolated vertices. Then G has an Euler circuit if and only if G is connected and all its vertices have even degree.

Therefore, the Konigsberg bridge problem has no solution.

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Proof(1)() Trivial.() Step 1. Start from an arbitrary

vertex, go through an unused edge whenever it exists, and terminate at a vertex whose incident edges are all used.

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5

6

4

2

3 1265461

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Proof(2)

Step 2. Remove used edges and repeat step 1. for the remaining graph (start from a visited vertex).

1

5

6

4

2

3 2342

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Proof(3)

Step 3. Augment the path obtained so far with the result of step 2.

1234265461

Step 4. Repeat step 2 and step 3 until all the edges are used. #

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Planar graphsA graph G is planar if it can be drawn in the plane

with its edges intersecting only at vertices of G; otherwise it is nonplanar.

ab

c d

a planar graph a nonplanar graph

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Elementary subdivisionLet G = (V,E) be a loop-free undirected graph,

where E. An elementary subdivision of G results when an edge e = (u,w) is removed from G and then the edges (u,v), (v,w) are added to G-e, where vV.

v

wu

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Homeomorphic graphsThe loop-free undirected graphs G1 and G2 are

called homeomorphic if they are isomorphic or if they can both be obtained from the same loop-free undirected graph H by a sequence of elementary subdivisions.

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Kuratowski’s TheoremA graph is nonplanar if and only if it contain a

subgraph that is homeomorphic to either K5 or K3,3.

Peterson graph

a

b

cd

ef

g

hi

j

j a d

bfe

c

g

h

i

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Euler’s Theorem

Theorem 11.6 Let G = (V,E) be a connected planar graph with |V| =v and |E| = e. Let r be the number of regions in the plane determined by a planar embedding. Then v – e + r = 2.

Proof. By induction.

v = 4e = 6r = 4 v – e + r = 2.

R1

R2

R3

R4

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Variant of Euler’s TheoremCorollary 11.3 Let G = (V,E) be a loop-free connected planar

graph with |V| =v and |E| = e > 2, and r regions. Then 3r 2e and e 3v - 6.

Proof. For each region R, the degree of R, denoted by deg(R), is the number of edges traversed in a closed walk about the boundary of R.

Each region has at least three edges.

Each region has degree 3.

The sum of the degrees of the r regions is 2e.

2e 3r. e 3v – 6. (By Euler’s Theorem) #

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Hamilton’s toyWilliam Rowan Hamilton

Irish mathematician, 1805-1865 In 1859, he developed a game sold to a Dublin toy

manufacturer. The object of the game was to find a cycle containing all vertices.

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Hamilton paths and cyclesIf G = (V,E) is a graph with |V| 3, we say that G has a Hamilton

cycle if there is a cycle in G that contains every vertex in V.

Theorem 11.8 Let G = (V,E) be a loop-free graph with |v| = n 2. If deg(x) + deg(y) n – 1 for all x, y V, x y, then G has a Hamilton path.

A Hamilton path is a path in G that contains each vertex.

Theorem 11.9 Let G = (V,E) be a loop-free graph with |v| = n 3. If deg(x) + deg(y) n – 1 for all nonadjacent x, y

V, then G has a Hamilton cycle.

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Gray code

A Gray code is a sequence s1, s2,…, such that every n-bit string appears somewhere in the

sequence sk and sk+1 differ in exactly one bit

and s1 differ in exactly one bit.

000001011010110111101 100

ns2

ns2

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A Hamiltonian cycle on the hypercube

010000

001

100

111

011

110

101

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Trees and ForestsLet G = (V,E) be a loop-free undirected graph.

The graph G is called tree if G is connected and contains no cycles.

G is called forest if G contains no cycles and is not connected.

treeforest

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Properties of TreesTheorem 12.1 If a, b are distinct vertices in a tree

T = (V,E), then there is a unique path that connects a and b.

Proof. By contradiction.

Theorem 12.3 In any tree T = (V,E), |V| = |E| + 1.

Proof. By Mathematical Induction on |E|.

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Properties of Trees (cont.)Theorem 12.4 For every tree T = (V,E), if |V| 2, then T

has at least 2 vertices of degree 1.

Proof. From Theorem 12.3 on

Theorem 12.5 The following statements are equivalent for a loop-free undirected graph G = (V,E).

(a) G is a tree.

(b) G is connected, but the removal of any edge from G disconnects G into two subgraphs that are trees.

(c) G contains no cycles, and |V| = |E| + 1.

(d) G is connected, and |V| = |E| + 1.

Vv

Ev .||2)deg(

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Brainstorm將 1~9數字填入九個圓圈中，相鄰的數字（有邊連結）之差不能小於 3 。數字不能重複，題目已經預填兩個數。

8

6

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Brainstorm根據已故的麻省理工 ( MIT )哲學及邏輯家 George Boolos，

以下的趣味邏輯問題可算是全世界最難的一個。有甲、乙、丙三位精靈，其中一位只說真話，另外一位只說假

話，還有一位隨機地決定何時說真話，何時說假話。你可以向這三位精靈發問三個是非題，而你的任務是從他們的答案找出誰說真話，誰說假話，誰是隨機答話。你每次可選擇任何一位精靈問話，問的問題可以取決於上一題的答案。這個難題困難的地方是這些精靈會以「 Da」或「 Ja」回答，但你並不知道它們的意思，只知道其中一個字代表「對」，另外一個字代表「錯」。你應該問那三個問題呢？

答案見 Boolos (1998) Logic, Logic, and Logic Harvard University Press, 弟 29章 .