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

sablay-logo

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

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

Coloring and Planarity

Graph Theory

sablay-logo

1

Coloring and Planarity Planar Graphs Coloring

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

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

sablay-logo

1

!"

4

Figure 9.5 An Electric Circuit with a Wire Crossing updilseal

2 Discrete Mathematics for Computer Science

CS 30

dcs-logo

acl-logo

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

sablay-logo

!"

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

CS 30

2

dcs-logo

acl-logo

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

sablay-logo

!"

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

CS 30

2

dcs-logo

acl-logo

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

sablay-logo

1

4 3

!"

Figure 9.6 An Electric Circuit Without Wire Crossings updilseal

dcs-logo

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

CS 30

acl-logo

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

!" ! "

sablay-logo

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.

Discrete Mathematics for Computer Science

CS 30

acl-logo

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.

sablay-logo

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

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.

sablay-logo

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

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

Planar Graphs Coloring

Coloring and Planarity

of Planar Figure 9.5 contains a battery Embedding of wires not marked by a node â€˘ 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

sablay-logo

4

4 updilseal

3

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

CS 30

dcs-logo

â– acl-logo

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

sablay-logo

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

dcs-logo

awn with one crossing. Discrete Mathematics for Computer Science

CS 30

acl-logo

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

sablay-logo

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

updilseal

dcs-logo

awn with one crossing. Discrete Mathematics for Computer Science

CS 30

acl-logo

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

sablay-logo

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

updilseal

dcs-logo

awn with one crossing. Discrete Mathematics for Computer Science

CS 30

acl-logo

Coloring and Planarity

Planar Graphs Coloring

one crossing.

K5 , with one crossing

sablay-logo

updilseal

dcs-logo

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

CS 30

acl-logo

Coloring and Planarity

Planar Graphs Coloring

How about K3,3 ?

CHAPTER 9

Water

â–

529

Graph Properties

h1

sablay-logo

Gas

W

h1

G

h2

E

h3

h2

h3 Electric

updilseal

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

CS 30

dcs-logo

â– acl-logo

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

sablay-logo

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

CS 30

dcs-logo

■ acl-logo

Coloring and Planarity

Planar Graphs Coloring

Regions of a Planar Embedding

f a graph is pictured.

sablay-logo

â–

updilseal

dcs-logo

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

CS 30

acl-logo

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

sablay-logo

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–

updilseal

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 .

dcs-logo

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

CS 30

acl-logo

Coloring and Planarity

Planar Graphs Coloring

sablay-logo

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

Coloring and Planarity

Planar Graphs Coloring

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

= (V , E ),

we have

|V | âˆ’ | E | + |R | = 2

sablay-logo

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

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

sablay-logo

Corollary. Given any planar simple graph G

= (V , E )

with |V |

≥ 3,

we have

|E | ≤ 3|V | − 6

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

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

sablay-logo

Corollary. Given any planar simple graph G

= (V , E )

with |V |

≥ 3,

we have

|E | ≤ 3|V | − 6 K5 is not planar!

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

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

sablay-logo

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

updilseal

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

|E | ≤ 2|V | − 4

Discrete Mathematics for Computer Science

CS 30

dcs-logo

acl-logo

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

sablay-logo

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

updilseal

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

|E | ≤ 2|V | − 4

dcs-logo

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

CS 30

acl-logo

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.

sablay-logo

B E

A

D

F

O updilseal

C

dcs-logo

unique unbounded region. Discrete Mathematics for Computer Science

â– CS 30

acl-logo

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

sablay-logo

B E

A

D

F

O updilseal

C

dcs-logo

unique unbounded region.

Figure 9.8 A Graph and Its Dual ■

Note that each dotted edge crosses exactly one solid edge,

Discrete Mathematics for Computer Science

CS 30

acl-logo

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

sablay-logo

updilseal

â– dcs-logo

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

CS 30

acl-logo

Coloring and Planarity

Planar Graphs Coloring

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

sablay-logo

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

wskiâ€™s Theorem

updilseal

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

Discrete Mathematics for Computer Science

CS 30

acl-logo

Coloring and Planarity

Planar Graphs Coloring

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

sablay-logo

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 (1896â€“1980), characterizes planar graphs in terms of two ki n subgraphs. 5

3 3

wskiâ€™s Theorem

updilseal

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

Discrete Mathematics for Computer Science

CS 30

acl-logo

Planar Graphs Coloring

Coloring and Planarity

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

sablay-logo

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

Discrete Mathematics for Computer Science

e CS 30

f

acl-logo

Planar Graphs Coloring

Coloring and Planarity

Another Example

1

6 2

sablay-logo

8

5

3

4

7 updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

Planar Graphs Coloring

Coloring and Planarity

Another Example

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

6 2

sablay-logo

1 8

2

5

5

1

3

6 4

3

5

8

7

2 updilseal

4

7 Discrete Mathematics for Computer Science

6

CS 30

7

3

dcs-logo

acl-logo

Coloring and Planarity

Planar Graphs Coloring

Crossing Number

sablay-logo

The crossing number of a graph G , denoted

Î˝(G ),

is the minimum

possible number of crossings in a drawing of G .

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

Coloring and Planarity

Planar Graphs Coloring

sablay-logo

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

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

âˆˆ Z+

sablay-logo

, 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

Ď‡(G ),

is the

minimum possible number of colors in a coloring of G .

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

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

âˆˆ Z+

sablay-logo

, 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

Ď‡(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. dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

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

âˆˆ Z+

sablay-logo

, 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

Ď‡(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. dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

Coloring and Planarity

Planar Graphs Coloring

âˆˆ Z+

sablay-logo

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

Ď‡(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

Discrete Mathematics for Computer Science

Ď‡(G ) â‰¤ k .

CS 30

dcs-logo

acl-logo

Coloring and Planarity

s

Planar Graphs Coloring

t

u

v

1

sablay-logo

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

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

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

sablay-logo

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

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

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

sablay-logo

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

Discrete Mathematics for Computer Science

dcs-logo

CS 30

acl-logo

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

Ď‰(G ),

is the maximum

number of vertices in a clique of G .

sablay-logo

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

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

Ď‰(G ),

is the maximum

number of vertices in a clique of G .

sablay-logo

u

v

w

x

y

z

updilseal

dcs-logo

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

CS 30

acl-logo

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 ).

sablay-logo

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

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

sablay-logo

x

z

y

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

dcs-logo

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

CS 30

acl-logo

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.

sablay-logo

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

CS 30

acl-logo

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

sablay-logo

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

2 acl-logo

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.

sablay-logo

v1

v6

v1

v2

v4

v7

v8

v5

v3

v2

v6

v4

updilseal

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

Discrete Mathematics for Computer Science

CS 30

acl-logo

Coloring and Planarity

Planar Graphs Coloring

Coloring Maps

D

C

E

B A

F

sablay-logo

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

Discrete Mathematics for Computer Science

CS 30

updilseal

dcs-logo

acl-logo

Coloring and Planarity

Coloring Maps

Planar Graphs Coloring

Figure 9.9 South America

2 1

sablay-logo

2 4 2 3 updilseal

1 dcs-logo

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

CS 30

acl-logo

merica

Coloring and Planarity

Planar Graphs Coloring

Coloring Maps

2

1 2 1

1 2

sablay-logo

3 4 2 4

3

updilseal

1

ph for South America

2

Discrete Mathematics for Computer Science

dcs-logo

CS 30

acl-logo

Coloring and Planarity

Planar Graphs Coloring

The Four Color Theorem

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

Ď‡(G ) â‰¤ 4.

sablay-logo

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

Coloring and Planarity

Planar Graphs Coloring

The Four Color Theorem

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

Ď‡(G ) â‰¤ 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

CS 30

sablay-logo

updilseal

dcs-logo

acl-logo

Coloring and Planarity

sablay-logo

Questions? See you next meeting!

updilseal

dcs-logo

Discrete Mathematics for Computer Science

CS 30

acl-logo

Read more

Similar to

Popular now

Just for you