122

CHAPTER 8

b-Colouring of Some Graph Operations

In this Chapter, Graph Operations like addition and deletion of a vertex, an edge in a

Cycle, Union and Complement of Path with Cycle, Strong Product of Path with Cycle,

Cartesian Product of Cycles, Corona Product of Path graph with Cycle, Path graph with

Complete graph, Cycle with Complete graph, Star graph with Complete graph are obtained.

The b-Chromatic number and Structural properties for above operations on graphs are

determined.

8.1 Introduction [57, 68,72, 76]

Operations on graphs produce new ones from older ones. There are several operations

on Graph Theory that produce new graphs from old ones, which is mentioned in the

following major categories.

• Unary Operations

• Binary Operations

Unary Operations

Unary operations create a new graph from the old one. Elementary operations are

sometimes called "editing operations" on graphs. They create a new graph from the original

one by a simple or a local change, such as addition or deletion of a vertex or an edge,

merging and splitting of vertices, edge contraction, etc. Advanced operations such as Line

graph, Dual graph, Complement graph, Graph minor, Power of graph, etc. also creates new

graphs from the old ones.

Binary Operations

Binary operations creates a new graph from two initial graphs G1(V1, E1) and G2(V2, E2):

• The disjoint union of graphs, sometimes referred as simply graph union, which is

defined as: For two graphs with disjoint vertex sets V1 and V2 and disjoint edge sets

E1 and E2, their disjoint union is the graph (V1∪V2, E1∪E2).

• The graph join of two graphs is their graph union with all the edges that connect the

vertices of the first graph with the vertices of the second graph.

123

• Graph products based on the Cartesian product of the vertex sets are:

Ø Cartesian product of graphs

Ø Lexicographic product of graphs also called Graph composition

Ø Strong product of graphs

Ø Tensor product of graphs, also called as direct product, categorical product,

cardinal product or Kronecker product.

Ø Zig-zag product of graphs

• Other graph operations called "products" are:

Ø Rooted product of graphs.

Ø Corona product or simply corona of G1 and G2

8.2 b-Chromatic Number of a Graph in Addition of Parallel Chords

8.2.1 Theorems

For any Cycle Cn, addition of parallel chords between non adjacent vertices holds the

following statements:

• When n is odd, there exists a unique 3 cycle and ���� times 4 cycle.

• When n is even, there exists more than one 3 cycle and ����- 2 times 4 cycle.

Proof

Let Cn be a Cycle with n vertices. Let v1,v2,….vn be the vertices and e1,e2,…ek be the

edges of the Cycle Cn with parallel chords, where k is defined as

� =���� � �� �������� � + 1�� �����

Case 1

When n is an odd cycle, it is clear that the edges e1,en,en+1 are mutually adjacent to

each other, so it obviously forms a 3 cycle. Also we see that the remaining number of vertices

in the Cycle is even. Under observation, it forms����times 4 cycles. Thus, when n is odd, there

exist a unique 3 cycle and ���� times 4 cycle.

124

Example

Figure 1: C7 with parallel chords

Case 2

When n is an even cycle, the vertices with minimum degree (i.e. δ =2) forms a 3 cycle

and vertices with maximum degree (i.e. ∆= 3) forms a 4 cycle. Thus for every even cycle

there exist more than one 3 cycle and ���� - 2 times 4 cycle.

Example

Figure 2: C10 with parallel chords

125

8.2.2 Theorem

The Cycle Cn with arallel chords has the b-Chromatic number four for every n≥8.

Example

Figure 3: C8 with parallel chords

8.3 b-Chromatic Number of a Graph when an Edge is removed

8.3.1 Theorem

For any Cycle (n ≥5) with an edge e∈V(Cn), φ(Cn) = φ(Cn–e)

Proof

Let Cn be the Cycle of length n. Let v1,v2,...,vn be the vertices arranged in

anticlockwise direction i.e. V(Cn)={v1,v2,....,vn} and the edge set be denoted as

E(Cn)={ e1,e2,e3,…en}. Here the vertex vi is adjacent with the vertices vi-1 and vi+1 for

i=2,3,…n-1, v1 is adjacent with v2, vn and the vertex vn is adjacent with vn-1 and v1. We know

that every Cycle is a connected graph with n vertices. From the Theorem 3.2.1, it is clear that

b-chromatic number of Cycle of length n for n≥5 is tricolourable i.e. φ(Cn) =3. Suppose if we

delete any edge from the Cycle, we obtain a Path graph of length n-1 which is again a

tricolourable graph under observation.

Therefore φ(Cn) = φ(Cn–e) for every n ≥5.

Example

Figure 4(a):φ(C8)=3 Figure 4(b): φ(C8-e)=3

126

8.3.2 Corollary

φ(Cn) ≠ φ(Cn–e) for every n ≤ 3.

Example

Figure 5(a): C3 Figure 5(b):C3-e

8.3.3 Corollary

For any Path for n ≥ 5, e∈V(Pn), φ(Pn) = φ(Pn–e)

8.4 b-Colouring of Adding a Pendant Vertex to each Vertex of a Cycle

8.4.1 Theorem

For any n ≥ 6, φ(Cn • K1) = φ(Wn)

Proof

Let vi for 1 ≤ i ≤ n are the vertices taken in the anticlockwise direction in the wheel

graph Wn, where vn is the hub. It is clear that the vertex vi is adjacent with the vertices vi-1

and vi+1 for i=2,3,…n-1, the vertex v1 is adjacent with v2 and vn-1, the vertex vn is adjacent with

all the vertices. Here every vertex except the hub is incident with three edges, so we assign

four colours, which produces a maximum and b-chromatic colouring by the colouring

procedure. Also we know that the b-chromatic number of any Cycle has three colours for

n ≥ 5. If we attach a pendant vertex to every vertex of Cycle Cn, it is obvious that it has four

colours under observation.

Therefore φ(Cn • K1) = φ(Wn)

127

Example

Figure 6(a): φ(C6 • K1) = 4 Figure 6(b): φ(W6 ) = 4

8.4.2 Results obtained by Removing Edges from the Complete Graph

• φ(K4 –e) = φ(C5) = φ(K1,n,n),(n ≥2)

• φ(K3 –e) = φ(C2)= φ(K1,n) ,(n ≥2)

• φ(K4 –2e) = φ(C3)

• φ(K5 –2e) = φ(Wn), (n ≥6)

• φ(K5 –3e) = φ(Cn), (n ≥5)

8.5 b-Chromatic Number of Union of Path with Cycle

The Disjoint union of graphs [31,45] sometimes referred as simply graph union,

which is defined as follows. Given two graphs G1 and G2, their union will be a graph such

that V(G1 ∪G2)= V(G1 ) ∪ V(G2) and E(G1 ∪G2)= E(G1) ∪ E(G2).

The Complement G′ [31,82] of a graph G is defined as a simple graph with the same

vertex set as G and where two vertices u and v are adjacent only when they are not adjacent

in G.

8.5.1 Theorem

For any Path Pn and the Cycle Cn with n vertices, the b-chromatic number of ������������� is given by φ��������������� = n-1 for n ≥2.

128

Proof

Let G1= Pn be a Path graph with n vertices and n-1 edges and G2 = Cn be a Cycle

with n vertices and n edges. Let G = G1∪G2 be the graph obtained by the union of subgraph

Pn and Cn of a graph has the vertex set V(Pn)∪ V(Cn) and edge set E(Pn)∪E(Cn).

Consider G = Pn∪Cn whose vertex set V(G) = {v1,v2,v3…v2n-2}. Here in Pn∪Cn, we see

that the vertex vi is adjacent with the vertices vi+1 and vi-1 for i=2,3…n-1,n+1,…2n-2, v1 is

adjacent with v2, v2n-2 and the vertex vn is adjacent with the vertices v1,vn-1 and vn+1.

Now consider the graph G=�������������� . By the definition of Complement, for any graph

G, the non-adjacent vertices are adjacent in its complement. Here �������������� contains 2n-2

vertices as in G1∪G2. Arrange the vertices of �������������� namely v1,v2,v3..vn,vn+1,…v2n-2 in

clockwise direction.

Assign a proper colouring to these vertices as follows. Consider the colour class

C={c1,c2,c3..cn-1}. First assign the colour ci to the vertex vi for i=1,2…2n-2, it will not

produce a b-chromatic colouring, due to the above mentioned non-adjacency condition.

Hence to make the colouring as b-chromatic one, assign the colour ���� ! � to the

vertices vi and vi+1 for i=1,3,5…2n-3. Now all the vertices vi for i=1,2…2n-2 realizes its own

colour, which produces a b-chromatic colouring. Furthermore it is the maximum colouring

possible.

Example

Figure 7(a): P5 ∪ C5

129

Figure 7(b):�"��"��������= 48.5.2 Theorem

φ (Pn∪Cn)= 3 for every n ≥ 3

Proof

The result is trivial from the above theorem.

8.5.3 Theorem

For any Path graph Pn and Cycle Cm with n and m vertices respectively, then

φ (Pn∪Cm) = 3 for n ≥ 2.

Example

Figure 8: P4∪C10 =3

130

8.5.4 Result

For any integer n>2, φ(Cn) = φ(�����) Example

Figure 9(a): φ(C6)=3 Figure 9(b): φ(�#���)=3

8.6 b-Chromatic Number of Strong Product of Path with Cycle

The strong product is also called the [3, 66, 89, 90] normal product and AND product.

It was first introduced by Sabidussi in 1960. In Graph theory, the strong product G⊗H of

graphs G and H is a graph such that the vertex set of G⊗H is the Cartesian product

V(G) × V(H); and any two vertices (u,u') and (v,v') are adjacent in G⊗H if and only if u' is

adjacent with v' or u'=v' and u is adjacent with v or u = v.

8.6.1 Theorem

If Pm is a Path graph on m vertices and Cn be a Cycle on n vertices respectively. Then

φ(Pm ⊗Cn) = 6, n ≥ 3 and m=2

Proof

By observation, the strong product of Pm⊗Cn is a 5-regular graph. Therefore we say

that φ[Pm⊗Cn] ≥ 5. By colouring procedure we assign six colours to every Pm⊗Cn, which

produces a b-chromatic colouring. Suppose if we assign more than six colours, it contradicts

the definition of b-colouring. Hence the b-chromatic number of Strong product of Pm⊗Cn is

six.

131

Example

Figure 10: φ (P2⊗ C5) = 6

8.6.2 Structural Properties of P2⊗⊗⊗⊗Cn

• Number of vertices in P2⊗Cn is two times the number of vertices in Cycle Cn.

• Number of edges in P2⊗Cn is five times the number of edges in Cycle Cn.

• Every P2⊗Cn is a 5 regular graph.

8.6.3 Theorem

If Pn is a Path graph of length n-1 and Km be a Complete graph on m vertices

respectively. Then φ(Pn⊗K2) =%4for * 36forevery 0 4 Example

Figure 11(a): φ(P2⊗K2)=4Figure 11(b): φ(P3⊗K2)=4

Figure 11(c): φ(P5⊗K2)=6

132

8.6.4 Theorem

Let Pn and Pm be the Path graph on n and m vertices respectively, then

φ(Pn⊗Pm) =% + 2for2 * * 467��8 = 9forevery 0 5 Proof

The Proof of the Theorem 8.6.3 and 8.6.4 is immediate from the Theorem 8.6.1.

Example

Figure12 (a): φ(P2⊗P2)=4 Figure12 (b): φ(P3⊗P3)=5

Figure 12 (c): φ(P5⊗P5)=9

133

8.7. b-Chromatic Number of Cartesian product of C3 with Cycle Cn

In Graph theory, the Cartesian product [75,76,89,90,91] G□H of graphs G and H is a

graph such that the vertex set of G□H is the Cartesian product V(G) × V(H); and any two

vertices (u,u') and (v,v') are adjacent in G□H if and only if either u = v and u' is adjacent with

v' in H, or u' = v' and u is adjacent with v in G.

8.7.1 Theorem

Let Cn and Cm be the Cycles on n and m vertices respectively, then φ (Cm□Cn) = 5 for

every n ≥ 6 and m=3.

Proof

By the colouring procedure under observation, we assign five colours to every Cm□Cn,

for producing a b-chromatic colouring. Suppose if we assign more than five colours, it

contradicts the definition of b-colouring. Thus by the colouring procedure, the b-chromatic

number of Cartesian product of Cm□Cn is five.

Example

Figure 13: φ(C3□C6) = 5

8.7.2 Corollary

φ (Cm□Cn) = n for every n ≤ 5,m=3

Example

Figure 14: φ(C3□C4) = 4

134

8.7.3 Theorem

Let Pn be a Path graph of length n and a Complete graph K2 respectively, then

φ(Pn□K2) = % for2 * * 44forevery 0 5

Example

Figure 15(a): φ(P3□K2)=3

Figure 15(b): φ(P5□K2)=4

8.7.4 Theorem

Let G be a Cartesian product of Pn□Pn for every n >1,

φ(Pn□Pn) = ; for = 24for = 3,45for 0 5 8.8 b-Chromatic Number of Corona Product of Graphs

In Graph Theory, Corona product[86,90] or simply Corona of G1 and G2, defined by

Frucht and Harary is the graph which is the disjoint union of one copy of G1 and |V1| copies

of G2 (|V1| is the number of vertices of G1) in which each vertex of the copy of G1 is

connected to all vertices of a separate copy of G2.

8.8.1 b-Chromatic Number of Corona Product of Path Graph with Cycle

8.8.2 Theorem

For any integer n >3, φ (Pn o Cn) = n

135

Proof

Let G1= Pn be a Path graph of length n-1 with vertices v1,v2,v3….vn and edges

e1,e2,e3….en-1. Consider G2 = Cn be a Cycle of length n whose vertices are v1,v2,v3….vn and

edges are denoted as e1,e2,e3….en.

Now consider the Corona product of G1 and G2 i.e. G= Pn o Cn is obtained by taking

unique copy of Pn with n vertices and n copies of Cn and joining the ith

vertex of Pn to every

vertex in ith

copy of Cn. The Vertex set is defined as follows:

i.e. V(G) =V(Pn)∪V(C1n)∪V(C

2n)∪V(C

3n)∪ …… . .∪ V(C

nn)

where V(Pn) = {v1,v2,v3….vn } and V(Cin) = ?@AB: 1 * � * , 1 * D * E

Now assign a proper colouring to these vertices as follows. Consider the colour class

C= {c1,c2,c3….cn}. First assign the colour ci to the vertex vi for i=1,2,3…n and assign the

colour to @AB as ci+j when i+j ≤ n and ci+j-n when i+j >n for 1≤ i ≤ n, 1≤ j ≤ n-1. Next the

only vertex remaining to be coloured is @AB for j=n. Suppose if we assign any new colour to @AB for i=1,2,3.n, j=n it will not produce a b-chromatic colouring, because @AB ( i=1,2,3..n,

j=n) is adjacent only with @A� and @A�F�. To make the above colouring as b-chromatic one, we

assign the colour to uji other than the colour which we assigned previously for @A�

and @A�F�. Now the verticesG�A: 1 * � * H realize its own colours, which produce a b-chromatic

colouring. Thus by the colouring procedure the above said colouring is maximum and

b-chromatic.

Example

Figure 16: φ(P4oC4) = 4

136

8.8.3 Structural Properties of Pn o Cn

The number of vertices in PnoCn (n >3) i.e. p(PnoCn)= n(n+1), number of edges in the

PnoCn i.e. q(Pn o Cn) =2n2+n-1. The maximum and minimum degree of PnoCn is denoted as

∆ = n+2 and δ = n-1 respectively. The number of vertices having maximum degree ∆ in

PnoCn is denoted by n(p∆)=n-2 and the number of vertices having minimum degree δ in

PnoCn is denoted by n(pδ)=n2 respectively.

8.8.4 Theorem

For any Path Pn and Cycle Cn, the number of edges in Corona product of Pn with

Cycle Cn is 2n2+n-1 i.e. q(PnoCn)= 2n

2+n-1

Proof

q(PnoCn) = Number of edges in largest subgraph + Number of edges not in any of

the largest subgraph

= n×(2n)+n-1

= 2n2+n-1

Therefore q(Pn o Cn)= 2n2+n-1

8.9 b-Chromatic number of Corona Product of Path with Complete Graph

8.9.1 Theorem

Let Pn and K2 be Paths and Complete graphs with n vertices respectively. Then

φ(PnoK2) = I + 1for = 2 for = 3and4 M 1for = 55forevery N 6 Example

Figure 17(a): φ(P3 o K2)= 3 Figure 17(b): φ(P4 o K2)= 4

137

Figure 17(c): φ(P7 o K2)= 5

8.9.2 Structural Properties of Pn o K2

The number of vertices in PnoK2 i.e. p(PnoK2)= 3n, number of edges in the PnoK2 i.e.

q(PnoK2) =3n+(n-1). The maximum and minimum degree of PnoK2 (n>3) is denoted as

∆ = 4 and δ = 3 respectively. The number of vertices having maximum degree in PnoK2 is

denoted by n(p∆)=n-2 and the number of vertices having minimum degree in PnoK2 is

denoted by n(pδ)=2 respectively.

8.9.3 Theorem

For any integer n, φ(PnoKn) = n+1

Proof

Let G1= Pn be a Path graph of length n-1 with n vertices and G2 = Kn be a complete

graph of n vertices.

Consider the Corona product of G1 and G2 i.e. G= PnoKn is obtained by taking unique

copy of Pn with n vertices and n copies of Kn and joining the ith

vertex of Pn to every vertex in

ith

copy of Kn. The vertex set of PnoKn is defined as follows:

i.e. V(G) = V(Pn) ∪ V(K1n)∪ V(K

2n) ∪ V(K

3n) ∪…… .∪ V(K

nn)

where V(Pn)={v1,v2,v3….vn} and V(Kin) = ?@AB: 1 * � * , ,1 * D * E.

By observation, we see that there are n copies of disjoint subgraph which induces a

clique of order n+1 (say Kn+1). Therefore we assign more than or equal to n+1 colours to

every corona product of path graph with complete graph. Consider the colour class

C={c1,c2,c3,c4…cn,cn+1}. Now assign a proper colouring to these vertices as follows. Suppose

if we assign more than n+1 colours, it contradicts the definition of b-chromatic coloring.

Thus to make the above colouring as b-chromatic one, we cannot asssign more

than n+1 colours i.e. φ(Pn o Kn) ≤ n+1. Therefore φ(Pn o Kn) = n+1. Thus by the colouring

procedure the above said coloring is maximum and b-chromatic colouring.

138

Example

Figure 18: φ(P4oK4) = 5

8.9.4 Structural Properties of Pn o Kn

The number of vertices in PnoKn i.e. p(PnoKn)= n(n+1), number of edges in the PnoKn

i.e. q(PnoKn) =��OP�!P��F�� �. The maximum and minimum degree of PnoKn is denoted as

∆ = n+2 and δ = n respectively. The number of vertices having maximum and minimum

degree in Pn o Kn is denoted by n(p∆)=n-2 and n(pδ)=n2 respectively.

8.9.5 Theorem

For any Path Pn and Complete graph graph Kn the number of edges in Corona product

of Pn with Kn is ��OP�!P��F�� � i.e. q(PnoKn) = ��OP�!P��F�� � Proof

q(Pno Kn) = Number of edges in all Kn+1+ Number of edges not in any of the Kn+1

= n×q(Kn+1)+ Number of edges not in any of the Kn+1

= n×Q�P�� R +n-1

= n��S�P�)� �+ n-1

= n��!P�� �+ n-1

= ��OP�!� � + n-1

= ��OP�!P��F�� � Therefore q(Pn o Kn) = ��OP�!P��F�� �

139

8.10 b-Chromatic Number of Corona Product of Fan Graph with K2

8.10.1 Theorem

φ(F1,noK2) =% + 1for2 * * 45forevery N 5 8.10.2 Theorem

φ(F1,noKn)= n+1 for every n>2.

Proof

The Proof of the theorem is similar to Theorem 8.6.1.

8.11 b-Chromatic Number of Corona Product of K1,n with Kn

8.11.1 Theorem

If K1,n and K2 are star graph and Complete graphs with n vertices respectively, then

φ(K1,n o K2) = T + 1for * 34for 0 4

8.11.2 Theorem

φ(K1,n o Kn) = n+1 for every n>2 .

Proof

The Proof of the theorem is similar to theorem (8.6.1)

8.12 b-Chromatic Number of Corona Product of Path Graph with K1

8.12.1 Theorem

For any n≥3, φ[Pn o (n-1) K1] = n

Proof

Let Pn be a Path graph of length n-1 i.e. V(Pn) = {v1,v2,v3..vn} and E(Pn)={e1,e2,..en-1}.

By the definition of Corona product, attach (n-1) copies of K1 to each vertex of Pn.

i.e. V [Pn o (n-1) K1] = G�A/1 * � * H ∪ ?�AB/1 * � * , 1 * D * M 1E. E[Cn o (n-3) K1] = G�A * � * M 1H ∪ ?�AB/1 * � * , 1 * D * M 1E

140

Consider the colour class C={c1,c2,c3..cn}. Assign a proper colouring to the vertices as

follows. Give the colour ci to vertex vi for i=1,2,3…n and assign the colour cn+1 to vij′s for

i=1,2…n and j=1,2,3..n-1. We see that each vertex vi is adjacent with vi-1 and vi+1 for

i=2,3,..n-1, v1 is adjacent with v2 and vn is adjacent with vn-1, due to this non-adjacency

condition of vi for i=1,2,3..n does not realize its own colour, which does not produce a

b-chromatic colouring. To make the colouring as b-chromatic one, assign the colouring to vi,

vij′s as follows. For 1 ≤ i ≤ n, assign the colour ci to vi. For i=1,2,3..n, j=1,2,3…n-1, assign

the colour ci+j to vij when i+j ≤ n and assign the colour ci+j-n when i+j>n. Now the vertices vi

for i=1,2,3..n realizes its own colour , which produces a b-chromatic colouring. Thus by the

colouring procedure the above said colouring is maximum and b-chromatic.

Example

Figure 19

Figure 20: φ [P4 o 3 K1] = 4

8.12.2 Structural Properties of Pn o (n-1) K1

The number of vertices in Pn o (n-1) K1 i.e. p[Pn o (n-1) K1]= n2, number of edges in the

Pn o (n-1) K1 i.e. q[Pn o (n-1) K1] =n2-1. The maximum and minimum degree of PnoKn is

denoted as ∆ = n+1 and δ = 1 respectively. The number of vertices with maximum and

minimum degree is n(p∆) = n+1, n(pδ)=n(n-1). Here n vertices with degree n+1 and n(n-1)

vertices with degree 1.

141

8.13 b-Chromatic Number of Corona Product of Cycle with K1

8.13.1 Theorem

For any n≥3, φ [Cn o (n-3) K1] = n

Proof

Let Cn be a Cycle of length n i.e. V(Cn) = {v1,v2,v3..vn} and E(Cn) = {e1,e2,e3..en}. By

the definition of Corona product, attach (n-3) copies of K1 to each vertex of Cn .

i.e. V [Cn o (n-3) K1] =G�A/1 * � * H ∪ ?�AB/1 * � * , 1 * D * M 3E E[Cn o (n-3) K1] = G�A/1 * � * H ∪ ?�AB/1 * � * , 1 * D * M 3E Here |V(G)| = n(n-2) and |E(G)| = n(n-2)

Consider the colour class C={c1,c2,c3..cn}. Assign a proper colouring to these vertices

as follows. Assign the colour ci to vertex vi for i=1,2,3…n and assign the colour cn+1 to the

vertex vij for i=1,2…n and j=1,2,3..n-3. Here the vertex vi for i=1,2,3..n does not realize its

own colour because each vi is adjacent with vi-1 and vi+1 for i=2,3,..n-1, v1 is adjacent with

v2,vn and vn is adjacent with vn-1 and v1. To make the colouring as b-chromatic, assign the

colouring to vi, vij’s as follows. For 1 ≤ i ≤ n, assign the colour ci to vi. For 1 ≤ i ≤ n ,

1 ≤ j ≤ n-3 assign the colour ci+j+1 to vij’s when i+j < n and assign the colour ci+j+1-n to

remaining vij′s when i+j ≥ n.

Now, here all the vertices vi for i=1,2,3..n realizes its own colour, which produces a

b-colouring. Thus by the colouring procedure the above said colouring is maximum and

b-chromatic.

Example

Figure 21: φ [C6 o 3K1] = 6

142

8.13.2 Structural Properties of Cn o (n-3) K1

The number of vertices and edges in Cno(n-3)K1 = n(n-2). The maximum and

minimum degree of Cno(n-3) K1 are denoted as ∆ = n-1 and δ=1 respectively. The number of

vertices having maximum degree in Cno(n-3) K1 is denoted by n(p∆)=n and the number of

vertices having minimum degree in Cno(n-3) K1 is denoted by n(pδ)=n(n-3) respectively.

8.14 b-Chromatic Number of Corona Product of Star Graph with Complete Graph

8.14.1 Theorem

For any n≥2, φ [K1,n o (n-1) K1] = n+1 where n ≥ 2,

Proof

Let v1,v2,v3….vn be the Pendant vertices of the graph K1,n with v as the root vertex. i.e.

V[K1,n] = G�H ∪ G�A/1 * � * H. Here v is adjacent with vi for i=1,2,3..n.

Now consider the vertex set of K1,n o (n-1) K1 is defined as follows:

i.e. V [ K1,n o (n-1) K1] = {v} ∪ G�A/1 * � * H ∪ ?�AB/1 * � * , 1 * D * M 1E ∪ G�AV/1 * � * M 1H.

Consider the colour class C={c1,c2,c3..cn,cn+1}. Assign a proper colouring to these

vertices as follows. Give the colour ci to vertex vi for i=1,2,3…n and assign the colour cn+1

to the root vertex v. Here the root vertex realizes its own colour but the vertices vi for

i=1,2,3..n does not realize its own colour, which does not produce a b-chromatic colouring.

Thus to make the colouring as b-chromatic one, assign the colouring as follows:

For 1 ≤ i ≤ n, assign the colour ci to vi. Assign the colour cn+1 to root vertex v. Next

assign proper colouring to the vij’s as follows. For i=1,2,3..n, j=1,2,3…n-1, assign the colour

ci+j to vij when i+j ≤ n and assign the colour ci+j-n when i+j>n. For the remaining vertex vi′

assign any colour ci for i=1,2..n-1.

Now, all the vertices vi for i=1,2,3..n and the root vertex v realizes its own colour,

which produces a b-chromatic colouring. Thus by the colouring procedure the above said

colouring is maximum and b-chromatic.

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Example

Figure 22: φ [K1,4 o (n-1) K1] = 5

8.14.2 Structural Properties of K1,no (n-1)K1

• Number of vertices in K1,no(n-1)K1 i.e p[K1,no(n-1)K1]= n(n+1).

• Number of edges in K1,no(n-1)K1 i.e. q[K1,no(n-1)K1] = n2+n-1.

• Maximum degree of K1,no(n-1)K1 is ∆ = 2n-1 .

• Minimum degree of K1,no(n-1)K1 is δ = 1 .

• The number of vertices having maximum degree ∆ in K1,no(n-1)K1 is n(p∆) =1.

• The number of vertices having minimum degree δ in K1,no(n-1)K1 is n(pδ)=n2-1 .

• n vertices with degree n.