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Discrete Math by R.S. Chang, Dept. CSIE, NDHU 2
An Introduction to Graph Theory
Chapter 11
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 3
Chapter 11 An Introduction to Graph Theory11.1 Definitions and Examples
Undirected graph Directed graph
isolated vertex
adjacent
loop
multipleedges
simple graph: an undirected graph without loop or multiple edges
degree of a vertex: number of edges connected(indegree, outdegree)
G=(V,E)
For simple graphs, deg(v Eiv Vi
) | | 2
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 4
Chapter 11 An Introduction to Graph Theory11.1 Definitions and Examples
x ypath: no vertex can be repeated a-b-c-d-etrail: no edge can be repeat a-b-c-d-e-b-dwalk: no restriction a-b-d-a-b-c
closed if x=yclosed trail: circuit (a-b-c-d-b-e-d-a, one draw without lifting pen)closed path: cycle (a-b-c-d-a)
a
b
c
d
e
length: number of edges inthis (path,trail,walk)
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 5
Chapter 11 An Introduction to Graph Theory11.1 Definitions and Examples
Theorem 1.1 Let = ( , ) be an undirected graph, with , , . If there exists a trail from to , then there is
a path from to .
G V Ea b V a b a b
a b
a x b
remove any cycle on the repeatedvertices
Def 11.4 Let G=(V,E) be an undirected graph. We call G connected if there is a path between any two distinct vertices of G.
ab
cd
e ab
cd
e
disconnected withtwo components
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 6
Chapter 11 An Introduction to Graph Theory11.1 Definitions and Examples
Def. 11.5 For any graph = ( , ), the number of componentsof is denoted by ( ). 1 ( ) | |
Can you think of an algorithm to determine ( )?
G V EG G
G V
G
Def. 11.6multigraph of multiplicity 3
multigraphs
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 7
Chapter 11 An Introduction to Graph Theory11.2 Subgraphs, Complements, and Graph Isomorphism
Def. 11.7 If = ( , ) is a graph, then is calleda subgraph of if and where each edge ofin is incident with vertices in
11 1
G V E G V EG V V E E
E V
1 1 11
( , ),
.
a
b
c
d
e
a
b
c
d
e
b
c
d
ea
cd
spanning subgraph V1=V
induced subgraphinclude all edges of E in V1
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 8
Chapter 11 An Introduction to Graph Theory11.2 Subgraphs, Complements, and Graph Isomorphism
Def. 11.11 complete graph: Kn
a
b
c
d
e
K5
Def. 11.12 complement of a graph
G Ga
b
c
d
e
a
b
c
d
e
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 9
Chapter 11 An Introduction to Graph Theory11.2 Subgraphs, Complements, and Graph Isomorphism
Theorem: Any graph of six vertices contains a K3 or K3.(In a party of six, There exists 3 people who are eithermutually acquainted or mutually inacquainted.)
5 is not enough.
a
b
c
d
e
For 6 people, let's look from the point ofview of a:
From the pigeonhole principle, there are3 who know a or 3 who does not know a.
a
b c d
a
b c dK3 or K3. K3 or K3.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 10
Chapter 11 An Introduction to Graph Theory11.2 Subgraphs, Complements, and Graph Isomorphism
Ex. 11.7 Instant Insanity, 4 cubes, each of the six faces on acube is painted with one of the colors, red (R), white (W),blue (B), or Yellow (Y). The object is to place the cubes in acolumn of four such that all four colors appear on each of thefour sides of the column.
W R Y W
Y
B (1)
B B W Y
R
Y (2)
R B Y B
R
W (3)
W R B Y
W
W (4)
There are (3)(24)(24)(24)=41472 possibilities to consider.
the bottom cube 6 faces with 4 rotations
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 11
Chapter 11 An Introduction to Graph Theory11.2 Subgraphs, Complements, and Graph Isomorphism
W R Y W
Y
B (1)
B B W Y
R
Y (2)
R B Y B
R
W (3)
W R B Y
W
W (4)
R W
Y B
13
11
2
4 2
3
4
4 3 2
Each edge correspondsto a pair of opposite faces.
R W
Y B1
2
3
4
R W
Y B2
4
1
3
Y
BRW
B
WBY
W
RYR
R
YWB
(1) (2) (3) (4)
Consider the subgraph of opposite column.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 12
Chapter 11 An Introduction to Graph Theory11.2 Subgraphs, Complements, and Graph Isomorphism
Graph Isomorphism
1 2
3 4
a b
c
d
w x y z
Def. 11.13 Let and be two undirected graphs. A function : is called a graphisomorphism if (a) is one - to - one and onto and (b) forall , ( , ) if and only if ( ( ), ( ))When such a function exists, and are called isomorphic graohs.
12
G V E G V Ef V V
fa b V a b E f a f b E
G G
1 1 1 2 2 21 2
1 21
( , ) ( , )
, .
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 13
Chapter 11 An Introduction to Graph Theory11.2 Subgraphs, Complements, and Graph Isomorphism
Ex. 11.8 q r
wz xy
u t
v
a
b
cd
e f
gh
i
j
a-q c-u e-r g-x i-z b-v d-y f-w h-t j-s, isomorphic
Ex. 11.9
degree 2 vertices=2
degree 2vertices=3
Can you think of an algorithm for testing isomorphism?
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 14
Chapter 11 An Introduction to Graph Theory11.3 Vertex Degree: Euler Trails and Circuits
degree 1 vertex: pendant vertex
For simple graphs, deg(v Eiv Vi
) | | 2Theorem 11.2
Corollary 11.1 The number of vertices of odd degree must be even.
Ex. 11.11 a regular graph: each vertex has the same degreeIs it possible to have a 4-regular graph with 10 edges?
2|E|=4|V|=20, |V|=5 possible (K5)
with 15 edges?
2|E|=4|V|=30 not possible
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 15
11.3 Vertex Degree: Euler Trails and Circuits
Chapter 11 An Introduction to Graph Theory
Ex. 11.12 The Seven Bridge of Konigsberg
area a
area b area d
area ca
b
c
d
Find a way to walk about the city so as to crosseach bridge exactly once and then return to thestarting point.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 16
11.3 Vertex Degree: Euler Trails and Circuits
Chapter 11 An Introduction to Graph Theory
Def. 11.15 Let G=(V,E) be an undirected graph or multigraphwith no isolated vertices. Then G is said to have an Euler circuitif there is a circuit in G that traverses every edge of the graph exactly once. If there is an open trail from a to b in G and thistrail traverses each edge in G exactly once, the trail is called anEuler trail.Theorem 11.3 Let G=(V,E) be an undirected graph or multigraphwith no isolated vertices. Then G has an Euler circuit if and onlyif G is connected and every vertex in G has even degree.
a
b
c
dAll degrees are odd. Hence no Euler circuitfor the Konigsberg bridges problem.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 17
11.3 Vertex Degree: Euler Trails and Circuits
Chapter 11 An Introduction to Graph Theory
proof of Euler circuit theorem:
Euler circult connected and even degree
v for other vertices
s for starting vertexobvious
connected and even degree Euler circuit
by induction on the number of edges.
e=1 or 2 e=n find any circuit containing s
s
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 18
11.3 Vertex Degree: Euler Trails and Circuits
Chapter 11 An Introduction to Graph Theory
Corollary 11.2 An Euler trail exists in G if and only if G isconnected and has exactly two vertices of odd degree.
two odd degree verticesa b
add an edge
Theorem 11.4 A directed Euler circuit exists in G if and only ifG is connected and in-degree(v)=out-degree(v) for all vertices v.
one in, one out
Can you think of an algorithm to construct an Euler circuit?
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 19
11.3 Vertex Degree: Euler Trails and Circuits
Chapter 11 An Introduction to Graph Theory
Ex. 11.13 Complete Cycles (DeBruijn Sequences)If n is a positive integer and N=2n, a cycle of length N of 0's and 1's is called a complete cycle if all possible subsequences of 0's and 1's oflength n appear in this cycle. n=1 01,
n=2 0011,n=3 00010111,00011101n=4 16 complete cyclesIn general
For n=3:
00
01 10
11
vertex set={00,01,10,11}a directed edge from x1x2 to x2 x3
a
b
c
d
e
f
g
h
Find an Euler circuit:
00111010abgfcdeh 00101110abcdefgh
2 2 1n n
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 20
Chapter 11 An Introduction to Graph Theory11.4 Planar Graphs
Def. 11.17 A graph (or multigraph) G is called planar if G can bedrawn in the plane with its edges intersecting only at vertices of G.Such a drawing of G is called an embedding of G in the plane.
Ex. 11.14,11.15 K1,K2,K3,K4 are planar, Kn for n>4 are nonplanar.
K4 K5
applications: VLSI routing, plumbing,...
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 21
11.4 Planar Graphs
Chapter 11 An Introduction to Graph Theory
Def. 11.18 bipartite graph and complete bipartite graphs (Km,n)
K4,4
K3,3 is not planar.Therefore, any graph containing K5
or K4,4 is nonplanar.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 22
11.4 Planar Graphs
Chapter 11 An Introduction to Graph Theory
Def. 11.19 elementary subdivision (homeomorphic operation)
u w u v w
G1 and G2 are called homeomorphic if they are isomorphicor if they can both be obtained from the same loop-free undirected graph H by a sequence of elementary subdivisions.
a b
cde
a b
cde
a b
cde
a b
cde
Two homeomorphic graphs are simultaneously planar or nonplanar.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 23
11.4 Planar Graphs
Chapter 11 An Introduction to Graph Theory
Theorem 11.5 (Kuratowski's Theorem) A graph is planar ifand only if it contains a subgraph that is homeomorphic toeither K5 or K3,3.
Ex. 11.17 Petersen graph
a
b
cd
e f
gh
i
j
a subgraph homeomorphic to K3,3
j ad
e f b
g
h
ci
Petersen graph is nonplanar.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 24
11.4 Planar Graphs
Chapter 11 An Introduction to Graph Theory
K4
R1R2
R3
R4
A planar graph divides the planeinto several regions (faces), one of them is the infinite region.
Theorem 11.6 (Euler's planar graph theorem)
For a connected planar graph or multigraph: v-e+r=2
numberof vertices
numberof edges
numberof regions
v=4,e=6,r=4, v-e+r=2
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 25
11.4 Planar Graphs
Chapter 11 An Introduction to Graph Theory
proof: The proof is by induction on e.
e=0 or 1 v=1r=1e=0
v=1r=2e=1
v=2r=1e=1
v-e+r=2
Assume that the result is true for any connected planar graph ormultigraph with e edges, where 0 e k
Now for G=(V,E) with |E|=k+1 edges, let H=G-(a,b) for a,b in V.
Since H has k edges, v e rH H H 2
And, v v e eG H G H , .1
Now consider the situation about regions.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 26
11.4 Planar Graphs
Chapter 11 An Introduction to Graph Theory
case 1: H is connected
a(=b) a(=b) a
ba
b
v e r v e r v e rG G G H H H H H H( ) ( )1 1 2
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 27
11.4 Planar Graphs
Chapter 11 An Introduction to Graph Theory
case 2: H is disconnected
a
b
a b
a
b
H1
H2
a bH1H2
v v v e e e r r rv e r
v e r v e r v ve e r r v e r
v e r
H H G H H G H H G
H H H
H H H G G G H H
H H H H H H H
H H H
1 2 1 2 1 2
1 1 1
2 2 2 1 2
1 2 1 2 1 1 1
2 2 2
1 12
21 1
2 2 2 2 2
, , .,
. ( )( ) ( ) ( )
( )
And by the induction hypothesis, Therefore,
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 28
11.4 Planar Graphs
Chapter 11 An Introduction to Graph Theory
degree of a region (deg(R)): the number of edges traversed in a shortest closed walk about the boundary of R.
R1
R2
R3
R4 R5
R6
R7
R8
two different embeddings
deg(R1)=5,deg(R2)=3deg(R3)=3,deg(R4)=7
deg(R5)=4,deg(R6)=3deg(R7)=5,deg(R8)=6
deg( ) deg( ) | |R R Eii
ii
1
4
5
818 2 9 2
abghgfda
a b
c
d fg h
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 29
11.4 Planar Graphs
Chapter 11 An Introduction to Graph Theory
Corollary 11.3 Let = ( , ) be a loop - free connected planargraph with | |= , | |= > 2, and regions. Then and
- .Proof: Since is a loop - free and is not a multigraph, the boundary of each region (including the infinite region) containsat least three edges. Hence, each region has degree 3.Consequently, = | |= the sum of the degrees of the regions determined by and . From Euler' s theorem,
2 = - + - + so - , or - .
G V EV v E e r r e
e vG
e E rG e r
v e r v ee
ve
v e e v
3 23 6
2 22 3
2
3 36 3 3 6
,
Only a necessary condition, not sufficient.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 30
11.4 Planar Graphs
Chapter 11 An Introduction to Graph Theory
Ex. 11.18 For K5, e=10,v=5, 3v-6=9<10=e. Therefore,by Corollary 11.3, K5 is nonplanar.
Ex. 11.19 For K3,3, each region has at least 4 edges, hence4r 2e. If K3,3 is planar, r=e-v+2=9-6+2=5. So 20=4r 2e=18,a contradiction.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 31
11.4 Planar Graphs
Chapter 11 An Introduction to Graph Theory
A dual graph of a planar graph
a b
c
d
e f
g
1
23
456
1
6 5
4
2
3
An edge in G corresponds with an edge in Gd.
It is possible to have isomorphic graphs with respective duals thatare not isomorphic.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 32
11.4 Planar Graphs
Chapter 11 An Introduction to Graph Theory
Def. 11.20 cut-set: a subset of edges whose removal increasethe number of components
Ex. 11.21
a
b
c
d
e
f
g
h
cut-sets: {(a,b),(a,c)},{(b,d),(c,d)},{(d,f)},...
a bridge
For planar graphs, cycles in one graph correspond to cut-setsin a dual graphs and vice versa.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 33
Chapter 11 An Introduction to Graph Theory11.5 Hamilton Paths and Cycles
a path or cycle that contain every vertex
Unlike Euler circuit, there is no knownnecessary and sufficient condition for a graph to be Hamiltonian.
Ex. 11.24 a b c
d e f
g hi
There is a Hamilton path, but no Hamilton cycle.
an NP-complete problem
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 34
Chapter 11 An Introduction to Graph Theory11.5 Hamilton Paths and Cycles
Ex. 11.25x
yy y y
x x
x
y
y
start labeling from here
4x's and 6y's, since x and y mustinterleave in a Hamilton path (or cycle),the graph is not Hamiltonian
The method works only for bipartite graphs.
The Hamilton path problem is still NP-complete when restrictedto bipartite graphs.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 35
Chapter 11 An Introduction to Graph Theory11.5 Hamilton Paths and Cycles
Ex. 11.26 17 students sit at a circular table, how many sittings arethere such that one has two different neighbors each time?
Consider K17, a Hamilton cycle in K17 corresponds to a seatingarrangements. Each cycle has 17 edges, so we can have (1/17)17(17-1)/2=8 different sittings.
12
3
4
5
6
17
16
15
1,2,3,4,5,6,...,17,1
12
3
4
5
6
17
16
15
1,3,5,2,7,4,...,17,14,16,1
12
3
4
5
6
17
16
15
1,5,7,3,9,2,...,16,12,14,1
14
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 36
Chapter 11 An Introduction to Graph Theory11.5 Hamilton Paths and Cycles
Theorem 11.7 Let be a complete directed graph, i.e. , has vertices and for any distinct pair , of vertices, exactlyone of the edges ( , ) or ( , ) is in Such a graph (called a
) always contains a directed Hamilton path.Proof: Let 2 with a path containing -1 edges ( If = , we' re finished. Ifnot, let be a vertex that doesn' t appear in
* *
*
K Kn x y
x y y x Ktournament
m p mv v v v v v m n
v p
n n
n
mm m
m
.
, ), ( , ), , ( , )..
1 2 2 3 1
case 1. v v1 v2 ...vm
case 2. v1 v2 ...vk v vk+1 ...vm
case 3. v1 v2 ...vm v
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 37
Chapter 11 An Introduction to Graph Theory11.5 Hamilton Paths and Cycles
Ex. 11.27 In a round-robin tournament each player plays everyother player exactly once. We want to somehow rank the playersaccording to the result of the tournament.
not always possible to have a ranking where a player in a certainposition has beaten all of the opponents in later positions
a b c
but by Theorem 11.7, it is possible to list the players such thateach has beaten the next player on the list
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 38
Chapter 11 An Introduction to Graph Theory11.5 Hamilton Paths and Cycles
Theorem 11.8 Let = ( , ) be a loop - free graph with | |= 2. If deg( ) + deg( ) -1 for all , , , then
has a Hamilton path.
G V EV n x y n x y V x yG
Proof: First prove that G is connected. If not,
x yn1 vertices n2 vertices
deg( ) deg( ) ( ) ( )x y n n n n n n 1 2 1 2 1 21 1 2 1
a contradiction
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 39
Chapter 11 An Introduction to Graph Theory11.5 Hamilton Paths and Cycles
Theorem 11.8 Let = ( , ) be a loop - free graph with | |= 2. If deg( ) + deg( ) -1 for all , , , then
has a Hamilton path.
G V EV n x y n x y V x yG
Assume a path pm with m vertices v1 v2 v3 ... vm
case 1. either v v1 or vm v
case 2. v1,v2,...,vm construct a cycle either v1 v2 v3 ... vm
or v1 v2 v3 ...vt-1 vt ... vm
otherwise assume deg(v1)=k, then deg(vm)<m-k.deg(v1)+deg(vm)<m<n-1, a contradiction
Therefore, v can be added to the cycle. v
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 40
Chapter 11 An Introduction to Graph Theory11.5 Hamilton Paths and Cycles
Corollary 11.4. If deg( )-
for all vertices, then the graph
has a Hamilton path.
Theorem 11.9 Let = ( , ) be a loop - free undirected graph with | |= 3. If deg( ) + ( ) for all nonadjacent
, , then contains contains a Hamilton cycle.
vn
G V EV n x y n
x y V G
1
2
deg
Proof: Assume G does not contain a Hamilton cycle. We add edges to G until we arrive a subgraph H of Kn where H has no Hamiltoncycle, but for any edge e not in H, H+e has a Hamilton cycle.
For vertices a,b wher (a,b) is not an edge of H. H+(a,b) has aHamilton cycle and (a,b) is part of it.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 41
Chapter 11 An Introduction to Graph Theory11.5 Hamilton Paths and Cycles
a(=v1) b(=v2) v3 ... vn
If (b,vi) is in H, then (a,vi-1) cannot be in H. Otherwise,b vi vn a vi-1 vi-2 v3 is a Hamilton cycle in H.
Consequently, deg which meansdeg a contradiction.
Corollary 11.5 If deg( ) for all vertices, then the graph has a
Hamilton cycle.
H HG G
a b na b n
vn
( ) deg ( ) ,( ) deg ( ) ,
2
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 42
Chapter 11 An Introduction to Graph Theory11.5 Hamilton Paths and Cycles
Corollary 11.6 If = ( , ) is a loop - free unirected graph with
| |= 3, and if | |-
then has a Hamilton cycle.
Proof: Let , where ( , ) . Remove all edges connectedeither to or and then , . Let = ( ' , ' ) denote the resultingsubgraph. Then | |=| ' |+ ( ) + ( ). Since | ' |= - ,
| ' |-
Consequently, -
| ' |+ ( ) + ( )
-
G V E
V n En
G
a b V a b Ea b a b H V E
E E a b V n
En n
E E a b
n
1
22
22
2
1
22
2
,
deg deg
. | | deg deg
21
22
2
2
+ ( ) + ( ). Therefore, ( ) + ( )
- -
and has a Hamilton cycle.
deg deg deg dega b a b
n nn G
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 43
Chapter 11 An Introduction to Graph Theory11.5 Hamilton Paths and Cycles
A related problem: the traveling salesman problem
a
b
c
d
e3
41
3
5 4
32
Find a Hamilton cycle of shortest total distance.
2 graph problem vs. Euclidean plane problem (computational geometry)
Certain geometry properties (for example, the triangle inequality) sometimes (but not always) make it simpler.
For example, a-b-e-c-d-a with total cost=1+3+4+2+2=12.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 44
Chapter 11 An Introduction to Graph Theory11.5 Hamilton Paths and Cycles
Two famous computational geometry problems.
1. closest pair problem: which two points are nearest2. convex hull problem
the convex hull
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 45
Chapter 11 An Introduction to Graph Theory11.6 Graph Coloring and Chromatic Polynomials
Def. 11.22 If G=(V,E) is an undirected graph, a proper coloringof G occurs when we color the vertices of G so that if (a,b) is anedge in G, then a and b are colored with different colors. Theminimum number of colors needed to properly color G is calledthe chromatic number of G and is written (G).
b
c
d
e
a3 colors are needed.a: Redb: Greenc: Redd: Bluee: Red
In general, it's a very difficult problem (NP-complete).
(Kn)=n
(bipartite graph)=2
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 46
Chapter 11 An Introduction to Graph Theory11.6 Graph Coloring and Chromatic Polynomials
A related problem: color the map where two regions arecolored with different colors if they have same boundaries.
G
R e
B
BR
Y
Four colors are enough for any map. Remaina mystery for a century. Proved with the aid of computer analysis in 1976.
a b
c
df
a
b
c
d
e
f
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 47
Chapter 11 An Introduction to Graph Theory11.6 Graph Coloring and Chromatic Polynomials
P(G,): the chromatic polynomial of G=the number of waysto color G with colors.
Ex. 11.31 (a) G=n isolated points, P(G,)=n.(b) G=Kn, P(G,)=(-1)(-2)...(-n+1)=(n)
(c) G=a path of n vertices, P(G,)=(-1)n-1.(d) If G is made up of components G1, G2, ..., Gk, then P(G,)=P(G1,)P(G2,)...P(Gk,).
Ex. 11.32e
G eG eG'
coalescing the vertices
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 48
Chapter 11 An Introduction to Graph Theory11.6 Graph Coloring and Chromatic Polynomials
Theorem 11.10 Decomposition Theorem for Chromatic Polynomials.If G=(V,E) is a connected graph and e is an edge, then P(Ge,)=P(G,)+P(G'e,).
e
G eG eG'
coalescing the vertices
a
b
In a proper coloring of Ge:case 1. a and b have the same color: a proper coloring of G'e case 2. a and b have different colors: a proper coloring of G.Hence, P(Ge,)=P(G,)+P(G'e,).
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 49
Chapter 11 An Introduction to Graph Theory11.6 Graph Coloring and Chromatic Polynomials
Ex. 11.33
e = -
P(Ge,)P(G,) P(G'e,)
P(G,)=(-1)3-(-1)(-2)=4-43+62-3Since P(G,1)=0 while P(G,2)=2>0, we know that (G)=2.
Ex. 11.34
= - = -2
e e
P(G,)=(4)-2(4)= (-1)(-2)2(-3) (G)=4
Title
• Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas.
• Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vivamus et magna. Fusce sed sem sed magna suscipit egestas.