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Coloring and Planarity

CS 30 : Discrete Mathematics for Computer Science First Semester, AY 2012-2013

https://sites.google.com/a/dcs.upd.edu.ph/nhsh_classes/cs30

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Nestine Hope S. Hernandez Algorithms and Complexity Laboratory Department of Computer Science University of the Philippines, Diliman nshernandez@dcs.upd.edu.ph updilseal

Day 22

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Discrete Mathematics for Computer Science

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Coloring and Planarity

Graph Theory

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1

Coloring and Planarity Planar Graphs Coloring

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is a graph that models this circuit, with edges representin An Electric with Wire Crossing graph can Circuit alternatively be drawn with no edge crossings, ht-hand side of Figure 9.6, the circuit can similarly be cons wire crossings, as shown on the left-hand side of Figure 9.6. Coloring and Planarity

Planar Graphs Coloring

3

2

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1

!"

4

Figure 9.5 An Electric Circuit with a Wire Crossing updilseal

2 Discrete Mathematics for Computer Science

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g, which we would like to avoid. On the right-hand side of is a graph that models this circuit, with edges representin raph that models this circuit, with edges representing wires. An Electric with Wire Crossing graph can Circuit alternatively be drawn with no edge crossings, h can alternatively be drawn with no edge crossings, as shown ht-hand of 9.6, Figure thecan circuit can similarly be cons nd side ofside Figure the 9.6, circuit similarly be constructed wire crossings, as shown on the left-hand side of Figure 9.6. ossings, as shown on the left-hand side of Figure 9.6. Coloring and Planarity

Planar Graphs Coloring

3

32

2

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!"

1

!"

41

4

ure Figure 9.5 An9.5 Electric Circuit with a Wire An Electric Circuit withCrossing a Wire Crossing updilseal

2 Discrete Mathematics for Computer Science

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2

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g, which we would like to avoid. On the right-hand side of is a graph that models this circuit, with edges representin raph that models this circuit, with edges representing wires. An Electric with Wire Crossing graph can Circuit alternatively be drawn with no edge crossings, h can alternatively be drawn with no edge crossings, as shown ht-hand of 9.6, Figure thecan circuit can similarly be cons nd side ofside Figure the 9.6, circuit similarly be constructed wire crossings, as shown on the left-hand side of Figure 9.6. ossings, as shown on the left-hand side of Figure 9.6. Coloring and Planarity

Planar Graphs Coloring

3

32

2

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!"

1

!"

41

4

Can this graph be drawn with no edge crossings?

ure Figure 9.5 An9.5 Electric Circuit with a Wire An Electric Circuit withCrossing a Wire Crossing updilseal

2 Discrete Mathematics for Computer Science

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2

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Coloring and Planarity

Planar Graphs Coloring

1

!" An Electric Circuit without Wire Crossing

4

Figure 9.5 An Electric Circuit with a Wire Crossing 2

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1

4 3

!"

Figure 9.6 An Electric Circuit Without Wire Crossings updilseal

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lications like Example 9.16 motivate our study of graphs t without crossings. Discrete Mathematics for Computer Science

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Coloring and Planarity

Planar Graphs Coloring

1

!" An Electric Circuit without Wire Crossing

1

!"

4

4

Figure 9.5 An Electric Circuit with a Wire Crossing

Figure 9.5 An Electric Circuit with a Wire Crossing

2

2

1 4

1 3

!" ! "

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4 3

Figure 9.6 9.6 An Electric CircuitCircuit WithoutWithout Wire Crossings â– Figure An Electric Wire Crossings updilseal

ations like Example 9.16 motivate our study of graphs that can be lications like Example 9.16 motivate our study of graphs t hout crossings. dcs-logo

without crossings.

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Coloring and Planarity

Planar Graphs Coloring

Planar Embedding

A planar embedding of a graph is a drawing of the graph such that the images of distinct edges do not intersect outside of their endpoints. That is, there are no crossings.

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Coloring and Planarity

Planar Graphs Coloring

Planar Embedding

A planar embedding of a graph is a drawing of the graph such that the images of distinct edges do not intersect outside of their endpoints. That is, there are no crossings.

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A graph is said to be planar if it has a planar embedding.

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Planar Graphs Coloring

Coloring and Planarity

of Planar Figure 9.5 contains a battery Embedding of wires not marked by a node • d. the right-hand side of ! On " A planar embedding of a graph1 is a drawing4of the graph such that the ith edges representing wires. images of distinct edges do not intersect outside of their endpoints. That th nois,edge crossings, shownCrossing there are no crossings. Electric Circuit withas a Wire can similarly be constructed A graph is said to be planar if it has a planar embedding. nd side of Figure 9.6. 2

!"

2

3

1 1

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4

4 updilseal

3

th a Wire Crossing ectric Circuit Without Wire Crossings Discrete Mathematics for Computer Science

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that a graph may have a drawing that is not a planar embedding an that the graph is not planar. Although the graph in Figure 9.5 Another Example with a crossing, the alternative drawing without crossings in Figs that it is planar. In Example 8.21, we saw two different drawings Coloring and Planarity

Planar Graphs Coloring

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rossing and one without crossings. The existence of the drawing sings implies that K 4 is planar. One might next try to find a planar of K 5 , but the result in Example 9.18 cannot be improved. updilseal

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awn with one crossing. Discrete Mathematics for Computer Science

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that a graph may have a drawing that is not a planar embedding an that the graph is not planar. Although the graph in Figure 9.5 Another Example with a crossing, the alternative drawing without crossings in Figs that it is planar. In Example 8.21, we saw two different drawings Coloring and Planarity

Planar Graphs Coloring

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K and is planar! rossing one without crossings. The existence of the drawing sings implies that K 4 is planar. One might next try to find a planar of K 5 , but the result in Example 9.18 cannot be improved. 4

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awn with one crossing. Discrete Mathematics for Computer Science

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that a graph may have a drawing that is not a planar embedding an that the graph is not planar. Although the graph in Figure 9.5 Another Example with a crossing, the alternative drawing without crossings in Figs that it is planar. In Example 8.21, we saw two different drawings Coloring and Planarity

Planar Graphs Coloring

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K and is planar! rossing one without crossings. The existence of the drawing How about K ? K is planar. One might next try to find a planar sings implies that 4 of K 5 , but the result in Example 9.18 cannot be improved. 4

5

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awn with one crossing. Discrete Mathematics for Computer Science

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Coloring and Planarity

Planar Graphs Coloring

one crossing.

K5 , with one crossing

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ng for a while and failing to find a planar emb Discrete Mathematics for Computer Science

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Coloring and Planarity

Planar Graphs Coloring

How about K3,3 ?

CHAPTER 9

Water

â–

529

Graph Properties

h1

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Gas

W

h1

G

h2

E

h3

h2

h3 Electric

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Figure 9.7 Connecting Three Utilities to Three Homes that the utility connections required here must accommodate at least one crossing. Discrete Mathematics for Computer Science

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Planar Graphs

529

C H A P T E R 9Coloring ■ Graph Properties

Coloring and Planarity

How about K3,3 ?

CHAPTER 9

529

Graph Properties

h1 Water

W

h1

Gas h2

W h2

h3

h1

G

G

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h1

h2

h2

h3

E

Electric

E

h3

h3 updilseal

Figure 9.7 Connecting Three to Three Homes 9.7 Connecting Three Utilities toUtilities Three Homes

that the utility connections required here must accommodate at least one nnections crossing. required here must accommodate at least one Discrete Mathematics for Computer Science

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Coloring and Planarity

Planar Graphs Coloring

Regions of a Planar Embedding

f a graph is pictured.

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â–

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ph may have a drawing that is not a planar embedding e graph is not planar. Although the graph in Figure 9.5 Discrete Mathematics for Computer Science

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Coloring and Planarity

Planar Graphs Coloring

0 Regions The ofregions the graph G pictured in Example 9.17 ar a Planar of Embedding f a graph is pictured. A, B, C, D, E, F, O.

B

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E

A

D

F

O

C The regions of the graph are labeled A, B , C , D , E , F , O where O is the

â–

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unique unbounded use RG ,unbounded or just R , to denote the set of Note that O isregion. the We unique region. regions of a graph G .

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ph may have a drawing that is not a planar embedding The standard way of drawing the dual of a graph b e graph is not planar. Although the graph in Figure 9.5 Discrete Mathematics for Computer Science

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Coloring and Planarity

Planar Graphs Coloring

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Coloring and Planarity

Planar Graphs Coloring

Euler's Formula Given any planar embedding of a connected graph G

= (V , E ),

we have

|V | − | E | + |R | = 2

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Coloring and Planarity

Planar Graphs Coloring

Euler's Formula Given any planar embedding of a connected graph G

= (V , E ),

we have

|V | − | E | + |R | = 2

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Corollary. Given any planar simple graph G

= (V , E )

with |V |

≥ 3,

we have

|E | ≤ 3|V | − 6

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Coloring and Planarity

Planar Graphs Coloring

Euler's Formula Given any planar embedding of a connected graph G

= (V , E ),

we have

|V | − | E | + |R | = 2

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Corollary. Given any planar simple graph G

= (V , E )

with |V |

≥ 3,

we have

|E | ≤ 3|V | − 6 K5 is not planar!

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Coloring and Planarity

Planar Graphs Coloring

Euler's Formula Given any planar embedding of a connected graph G

= (V , E ),

we have

|V | − | E | + |R | = 2

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Corollary. Given any planar simple graph G

= (V , E )

with |V |

≥ 3,

we have

|E | ≤ 3|V | − 6 K5 is not planar! Corollary. Given any planar simple graph G

= (V , E )

with |V |

≥3

and no triangles

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(that is, no 3-cycles), we have

|E | ≤ 2|V | − 4

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Coloring and Planarity

Planar Graphs Coloring

Euler's Formula Given any planar embedding of a connected graph G

= (V , E ),

we have

|V | − | E | + |R | = 2

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Corollary. Given any planar simple graph G

= (V , E )

with |V |

≥ 3,

we have

|E | ≤ 3|V | − 6 K5 is not planar! Corollary. Given any planar simple graph G

= (V , E )

with |V |

≥3

and no triangles

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(that is, no 3-cycles), we have

|E | ≤ 2|V | − 4

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K3,3 is not planar! Discrete Mathematics for Computer Science

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Planar Graphs Coloring

regions and the dual depend not only on the graph but also on Regions and Duals of awill Planar Embedding particular embedding always be fixed in context and is cted in the notation. In fact, the isomorphism type of the dual n the chosen embedding. (See Exercise 13.) Coloring and Planarity

The dual graph, denoted D (G ), is the graph with vertex set RG and edge

set EG for which the endpoints of each edge e are taken to be the regions

e graphthat, G pictured in Example are labeled here in the embedding, share 9.17 the image of e as part of their boundary. O.

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B E

A

D

F

O updilseal

C

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unique unbounded region. Discrete Mathematics for Computer Science

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Aregions R T II ■ and Combinatorics the dual

Planar Graphs Coloring

depend not only on the graph but also on Regions and Duals of awill Planar Embedding particular embedding always be fixed in context and is cted in the notation. In fact, the isomorphism type of the dual 9.21 embedding. We construct(See theExercise dual of the nPLE the chosen 13.)graph G from Example 9.17. In ◦ hasvertex beensetplaced The dualrepresented graph, denotedby D (an G ),open is the point graph with RG andinside edge each set EG for which the endpoints of regions each edgethat e are taken an to be the as regions the dotted lines join share edge part of th e graphthat, G pictured in Example are labeled here in the embedding, share 9.17 the image of e as part of their boundary. O. Coloring and Planarity

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B E

A

D

F

O updilseal

C

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unique unbounded region.

Figure 9.8 A Graph and Its Dual ■

Note that each dotted edge crosses exactly one solid edge,

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Note that each dotted edge crosses exactly one solid ed Regions and Duals of a Planar Embedding dual graph D(G) can then be drawn by itself in the usu vertices and edges. f a graph is pictured. Coloring and Planarity

Planar Graphs Coloring

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Note that, for any graph G with a planar embeddin âˆź Discrete Mathematics for Computer Science

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Coloring and Planarity

Planar Graphs Coloring

is Edge a sequence of edge subdivisions. Subdivisions and Kuratowski's Thm

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ollowing result, which we owe to the Polish mathematician C ski (1896–1980), characterizes planar graphs in terms of two ki n subgraphs.

wski’s Theorem

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is not planar if and only if it contains a subgraph that is a subdivision of eithe Equivalently, G is not planar if and only if G contains a subgraph homeomorp 3,3 . dcs-logo

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Coloring and Planarity

Planar Graphs Coloring

is Edge a sequence of edge subdivisions. Subdivisions and Kuratowski's Thm

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Kuratowski's Theorem

ollowing result, which weonly owe the aPolish mathematician C A graph is not planar if and if it to contains subgraph that is a subdivision of either K or K , . ski (1896–1980), characterizes planar graphs in terms of two ki n subgraphs. 5

3 3

wski’s Theorem

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is not planar if and only if it contains a subgraph that is a subdivision of eithe Equivalently, G is not planar if and only if G contains a subgraph homeomorp 3,3 . dcs-logo

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Planar Graphs Coloring

Coloring and Planarity

is Edge a sequence of edge subdivisions. Subdivisions and Kuratowski's Thm

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Kuratowski's Theorem

ollowing result, which weonly owe the aPolish mathematician C A graph is not planar if and if it to contains subgraph that is a subdivision of either K or K , . ski (1896–1980), characterizes planar graphs in terms of two ki CHAPTER 9 Graph Properties 533 n subgraphs. 5

3 3

The Petersen graph is not planar!

a

b

a

wski’s Theorem planart f

b r updilseal

if vandr

is not only if it contains a subgraph that is a subdivision of eithe c d c Equivalently, G is not planar if and only if G contains a subgraph homeomorp s t 3,3 . dcs-logo

s

e

d

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Planar Graphs Coloring

Coloring and Planarity

Another Example

1

6 2

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8

5

3

4

7 updilseal

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Planar Graphs Coloring

Coloring and Planarity

Another Example

d on the left below is a layout for a power grid for 1

6 2

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1 8

2

5

5

1

3

6 4

3

5

8

7

2 updilseal

4

7 Discrete Mathematics for Computer Science

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7

3

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Coloring and Planarity

Planar Graphs Coloring

Crossing Number

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The crossing number of a graph G , denoted

ν(G ),

is the minimum

possible number of crossings in a drawing of G .

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Coloring and Planarity

Planar Graphs Coloring

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Coloring and Planarity

Planar Graphs Coloring

Let G be a graph. (a) A coloring of G is an assignment of colors to the vertices of G in such a way that no two adjacent vertices have the same color. (b) A color class for a coloring is a set of all the vertices of one color. The vertices are partitioned by the color classes. (c) For any k

∈ Z+

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, a k -coloring of G is a coloring that uses

k dierent colors. (d) We say that G is k -colorable if there exists a coloring of G that uses at most k colors. (e) The chromatic number of G , denoted

χ(G ),

is the

minimum possible number of colors in a coloring of G .

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Coloring and Planarity

Planar Graphs Coloring

Let G be a graph. (a) A coloring of G is an assignment of colors to the vertices of G in such a way that no two adjacent vertices have the same color. (b) A color class for a coloring is a set of all the vertices of one color. The vertices are partitioned by the color classes. (c) For any k

∈ Z+

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, a k -coloring of G is a coloring that uses

k dierent colors. (d) We say that G is k -colorable if there exists a coloring of G that uses at most k colors. (e) The chromatic number of G , denoted

χ(G ),

is the

minimum possible number of colors in a coloring of G . It is not possible to color a graph containing loops.

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However, the existence of multiple edges has no impact on colorings. dcs-logo

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Coloring and Planarity

Planar Graphs Coloring

Let G be a graph. (a) A coloring of G is an assignment of colors to the vertices of G in such a way that no two adjacent vertices have the same color. (b) A color class for a coloring is a set of all the vertices of one color. The vertices are partitioned by the color classes. (c) For any k

∈ Z+

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, a k -coloring of G is a coloring that uses

k dierent colors. (d) We say that G is k -colorable if there exists a coloring of G that uses at most k colors. (e) The chromatic number of G , denoted

χ(G ),

is the

minimum possible number of colors in a coloring of G . It is not possible to color a graph containing loops.

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However, the existence of multiple edges has no impact on colorings. We typically use integers as the colors in our colorings. dcs-logo

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Coloring and Planarity

Planar Graphs Coloring

Let G be a graph. (a) A coloring of G is an assignment of colors to the vertices of G in such a way that no two adjacent vertices have the same color. (b) A color class for a coloring is a set of all the vertices of one color. The vertices are partitioned by the color classes. (c) For any k

∈ Z+

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, a k -coloring of G is a coloring that uses

k dierent colors. (d) We say that G is k -colorable if there exists a coloring of G that uses at most k colors. (e) The chromatic number of G , denoted

χ(G ),

is the

minimum possible number of colors in a coloring of G . It is not possible to color a graph containing loops.

updilseal

However, the existence of multiple edges has no impact on colorings. We typically use integers as the colors in our colorings. If a graph G has a k -coloring, then

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χ(G ) ≤ k .

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Coloring and Planarity

s

Planar Graphs Coloring

t

u

v

1

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w

x

y

z

4

The coloring shown in the middl right is a 4-coloring. Since G is th χ (G) ≤ 4. In fact, χ(G) = 4, as w on the right, the color classes are updilseal

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e to color a graph containing loops. However, the ltiple edges has no impact on colorings. Planar Graphs Coloring

Coloring and Planarity

s “colors”t in our colorings. u e integers as the s a k-coloring, then χ(G) ≤ k.

v

1

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below at the left, followed by two different colorings

w v

1

2

x 3

y 4

1

z 2

4 3

4

The coloring shown in the middl right is a 4-coloring. Since G is th 1 4. In 2 fact, 2 χ(G) 1 = 34, as w 1 χ5 (G) ≤ on the right, the color classes are updilseal

z

4

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oe color to color a graph a graph containing containing loops. loops. However, However, thethe ple ltiple edges edges hashas no no impact impact on on colorings. colorings. Coloring and Planarity

Planar Graphs Coloring

s“colors” u ntegers e integers as the as the “colors” int our in our colorings. colorings. sk-coloring, a k-coloring, then then χ(G) χ(G) ≤ k.≤ k.

v

1

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below ow at the at the left,left, followed followed by by twotwo different different colorings colorings

w v 1 1

2 2

x 3 3

y

4 4 1 1

z 2 2

3 3

4 4 4

The coloring shown in the middl right is a 4-coloring. Since G is th 1 ≤ 1 4.2 In 2 fact, 2 2 χ(G) 1 1 = 3 34, as 1 w 1 χ5 (G) on the right, the color classes are updilseal

z 4 4

5

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Coloring and Planarity

Planar Graphs Coloring

Let G be a graph. (a) A clique in G is a subgraph that is complete. (b) The clique number of G , denoted

ω(G ),

is the maximum

number of vertices in a clique of G .

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ue of G.

Planar Graphs Coloring

Coloring and Planarity

Let G be a graph.

raph G

(a) A clique in G is a subgraph that is complete. (b) The clique number of G , denoted

ω(G ),

is the maximum

number of vertices in a clique of G .

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u

v

w

x

y

z

updilseal

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he vertices v, w, and y form a clique of size 3, a more vertices. Discrete Mathematics for Computer Science

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Coloring and Planarity

Planar Graphs Coloring

A lower bound for the chromatic number Let G be any graph without loops. Then

χ(G ) ≥ ω(G ).

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Planar Graphs Coloring

Coloring and Planarity

A lower bound for the chromatic number

ed graph G

Let G be any graph without loops. Then

u

χ(G ) ≥ ω(G ).

v

w

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x

z

y

3. The vertices v, w, and y form a clique of size 3, and t 4 or more vertices. updilseal

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ue number provides a lower bound for the chromatic n Discrete Mathematics for Computer Science

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Planar Graphs Coloring

Coloring and Planarity

A lower bound for the chromatic number

ed graph G

9

Let G be any graph without loops. Then

4

u

8 χ(G 3 ) ≥ ω(G ).

v

w

ed the Gro·· tzsch graph and satisfies χ(G) = 4 > 2 = ω(G). A proof = 4 is left for the exercises.

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ommodating Scheduling Conflicts). In Example 8.3, a graph G

x

z

y

Astr.

Bio.

3. The vertices v, w, and y form a clique of size 3, and t 4 or more vertices. Fr. Calc. updilseal

Eng. Discr. ue number provides a lower bound for the chromatic n dcs-logo

used to reflect scheduling conflicts among classes in Astronomy, Bio Discrete Mathematics for Computer Science

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Coloring and Planarity

Planar Graphs Coloring

A Greedy Coloring Algorithm

Algorithm to color the pictured graph given the v6 , v7 of its vertices. v5

v3

v7

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v1

v2

v6

v4

updilseal

coloring is shown in the right-hand picture below. dcs-logo

2 Discrete Mathematics for Computer Science

1 CS 30

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an optimal coloring, as it did in Exam Coloring and Planarity

Planar Graphs Coloring

A Greedy Coloring Algorithm

Algorithm to color the pictured graph given the AMPLE The Greedy Coloring Algorithm app v6 , v7 of 9.32 its vertices. v5

v3

the ordering v1 , v2 , v3 , v4 , v5 , v6 , v7 , v8 at thev7 right.

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v1

v6

v1

v2

v4

v7

v8

v5

v3

v2

v6

v4

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coloring is shownEven in the right-hand picture below. though the vertices are colored

which 2is often1 a good 2 choice, the resu bipartite and thus has chromatic num dcs-logo

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Coloring and Planarity

Planar Graphs Coloring

Coloring Maps

D

C

E

B A

F

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G

M

A ! Guinea B ! Suriname C ! Guyana D ! Venezuela E ! Colombia F ! Equador G ! Peru H ! Bolivia I ! Chile J ! Paraguay K ! Argentina L ! Uruguay M ! Brazil

H J

I

K

L

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Coloring and Planarity

Coloring Maps

Planar Graphs Coloring

Figure 9.9 South America

2 1

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2 4 2 3 updilseal

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Figure 9.10 Coloring of Dual Graph for South America Discrete Mathematics for Computer Science

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merica

Coloring and Planarity

Planar Graphs Coloring

Coloring Maps

2

1 2 1

1 2

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3 4 2 4

3

updilseal

1

ph for South America

2

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Coloring and Planarity

Planar Graphs Coloring

The Four Color Theorem

The Four Color Theorem If G is any planar graph, then

χ(G ) ≤ 4.

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Coloring and Planarity

Planar Graphs Coloring

The Four Color Theorem

The Four Color Theorem If G is any planar graph, then

χ(G ) ≤ 4.

Although it was probably believed much earlier by map makers, the Four Color Theorem was rst formally conjectured in 1852 by an Englishman, Francis Guthrie(1831-1899). Twenty-seven years later, an erroneous proof was published by the English mathematician Arthur Kempe(1849-1922). The error was not caught until 1890, by another English mathematician, Percy Heawood (1861-1955). The Four Color Theorem was rst correctly proved in 1976 at the University of Illinois by the American mathematician Kenneth Appel (1932- ) and the German-born American mathematician Wolfgang Haken (1928-). Their proof required hundreds of pages of arguments, over 1200 hours of computer time, and ultimately the consideration of nearly 2000 cases. It was the rst computer-aided proof and was quite controversial at the time, since mathematicians could not check their argument by hand. However, their proof is now accepted, and many other computer-aided proofs of mathematical results have followed.

Discrete Mathematics for Computer Science

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