84 CHAPTER 6 b-Chromatic Number of Total Graph of Some Graphs In this Chapter, some special properties of Total graph of Cycle and Path are discussed along with its b-Chromatic number. Also the b-Chromatic number of Total graph of Star graph, Double Star graph, Fan graph, Bistar, Complete Bipartite graph, Crown graph are obtained along with its structural properties. 6.1 Introduction [2, 11, 84] Let G be a graph with vertex set V(G) and edge set E(G). Total graphs are generalizations of Line graphs. The Total Graph of graph G, denoted by T(G) is defined as follows. The vertex set of T(G) is V(G)∪ E(G). Two vertices x, y in the vertex set of T(G) are adjacent in T(G) in case one of the following condition holds: • x, y are in V(G) and x is adjacent to y in G. • x, y are in E(G) and x, y are adjacent in G • x is in V(G), y is in E(G) and x, y are incident in G. Example Figure 1(a): C 4 Figure 1(b): Total Graph of C 4
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84
CHAPTER 6
b-Chromatic Number of Total Graph of Some Graphs
In this Chapter, some special properties of Total graph of Cycle and Path are discussed
along with its b-Chromatic number. Also the b-Chromatic number of Total graph of Star graph,
Double Star graph, Fan graph, Bistar, Complete Bipartite graph, Crown graph are obtained along
with its structural properties.
6.1 Introduction [2, 11, 84]
Let G be a graph with vertex set V(G) and edge set E(G). Total graphs are generalizations
of Line graphs. The Total Graph of graph G, denoted by T(G) is defined as follows. The vertex
set of T(G) is V(G)∪ E(G). Two vertices x, y in the vertex set of T(G) are adjacent in T(G) in case
one of the following condition holds:
• x, y are in V(G) and x is adjacent to y in G.
• x, y are in E(G) and x, y are adjacent in G
• x is in V(G), y is in E(G) and x, y are incident in G.
Example
Figure 1(a): C4 Figure 1(b): Total Graph of C4
85
The following observations on Total Graph are made in [85]
• For any graph G with v>1 and e ≥1, the Total graph has atleast one 3 cycle.
• For all n-regular graph, Total graph is a 2n regular graph.
• For any graph G, and any vertex v in G, the degree of v in T(G) is twice the degree of
v in G.
• The number of edges in T(G) is equal to n times the number of vertices in T(G) if G
is a n-regular graph.
6.2 b-Chromatic Number of Total Graph of Cycle and its Structural Properties
6.2.1 Theorem
The b-Chromatic number of Total graph of every Cycle is 5 for every n≥5.
i.e. φ[T(Cn)] =5 for n ≥ 5.
Proof
Total graph of every Cycle is a 4-regular graph i.e. every vertex in the Total graph of
Cycle is incident with four edges. So we can assign more than or equal to five colours to the
Total graph of Cycle (for n≥5) for producing a b-chromatic colouring. Suppose if we assign
more than five colours, it contradicts the definition of b-colouring because in T(Cn) each vertex is
adjacent only with four vertices. Thus by the colouring procedure, the b-chromatic number of
Total graph of every Cycle is 5.
Example
Figure 2 : φ[ T(C5)] = 5
86
6.2.1.1 Remark
The Number of vertices in T(Cn) is twice the number of vertices in Cycle Cn.
6.2.1.2 Remark
The Number of edges in T(Cn) is four times the number of edges in Cycle Cn.
6.2.2 Theorem
The Total graph of every Cycle Cn (n>4) has 2n times 3-cycle and twice n cycles.
Proof
In T(Cn), by definition each edge vi,vi+1 is subdivided by the new vertex vi′ for i=1,2,3..n-1.
Consider the following cases to prove the above statement.
Case 1
Consider any arbitrary vertex vi. Here the vertex vi is adjacent vi+1, vi+1′,vn and vn′ for
i =2,3,4,..n-1 and vn is adjacent with vi-1, vi-1, v1 and v1′ and v1 is adjacent with v1′,v2,vn and vn′.
Here for i=2,3..n-1, vi along with vi′ and vi-1′ forms a 3 cycle and vi,vi+1 and vi’ forms another
3 cycle. Similarly the vertex v1 along with the vertices v1′ and vn′ forms a three cycle and the
vertex v1 along with v2 and v5 forms another 3 cycle, vertex vn with vn′ and v1 forms a three cycle
and the vertex Vn′ along with vn-1 and vn-1′ forms another 3 cycle. Thus, there are 2n times 3
cycles.
Case 2
Under observation , the vertices vi′ for i=1,2,3..n forms a n cycle and vi for i=1,2,3..n
forms another n cycle. Clearly there are two n cycles.
Example
Figure 3 : T (Cn)
87
6.2.2.1 Observation
The Total graph of Cycle T(C3) has (2n+1) times 3-cycle.
6.2.2.2 Observation
The Total graph of Cycle T(C2) has unique 3 cycle.
6.2.2.3 Remark
The Total graph of Cycle Cn is Eulerian and Hamiltonian.
6.3. b-Chromatic Number of Total Graph of Path and its Properties
6.3.1 Theorem
The b-Chromatic number of Total graph of every Path is 5 for every n ≥ 5.
Proof
Let Pn be a Path of length n-1 with vertices v1,v2,v3…vn. Let ui = vivi+1 for 1≤ i ≤ n-1 be
the edges of the Path Pn. By the definition of Total graph, V[T(Pn)]= V(Pn)∪E(Pn) i.e.
V[T(Pn)] = {vi,ui: 1≤ i ≤ n} and E[T(Pn)] = {vi, ui, uivi+1, vivi+1 ,uiui+1 : 1≤ i ≤ n-1}.
Consider any internal vertex ui or vi for i=2,3,4..n-1 in T(Pn) . Here the vertex vi or ui is
adjacent only with four vertices, so there is a possibility of assigning only five colours to the
Total graph of Path graph for every n ≥ 5, which produces a b-chromatic colouring. Suppose if
we assign more than five colours, it contradicts the definition of b-chromatic colouring. Thus by
the colouring procedure the above said colouring produces a maximum and b-chromatic
colouring.
Example
Figure 4: T(Pn)
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Figure 5: φ [ T(P6)] = 5.
6.3.2 Structural Properties of Total Graph of Path
• Number of vertices in T(Pn)= 2n-1
• Number of edges in T(Pn)= 4n-5
• Maximum degree of T(Pn) is ∆ = 4
• Minimum degree of T(Pn) is δ = 2
• n+1 vertices is of degree 4 and two vertices of degree 2 and remaining vertices is of
degree 3
6.3.3 Theorem
The Graph G is a Cycle if and only if G is T(P2).
6.4 b-Chromatic Number of Total Graph of Star Graph
6.4.1 Theorem
For n ≥ 3, ( )1, 1nT k nϕ = +
Proof
Consider the Star graph K1,n with V(K1,n) = v1, v2, v3, …., v where v is the root vertex. In
T(K1,n) by the definition of Total graph, each edge vvi for 1 ≤ i ≤ n of K1,n is subdivided by the
vertex v1′, v2′, …., vn′. The vertex set of Total graph of Star graph is defined as follows:
V [T(K1,n)] ={ vi / 1≤ i ≤ n } ∪{ vi′/ 1≤ i ≤ n }∪{v}.
Here the root vertex v along with v1′, v2′, …., vn′ induces a clique of order n+1 in T (K1,n)
Now assign a proper colouring to these vertices as follows. First assign the colours ci to the
vertex vi′ for 1≤ i ≤ n and assign the colour cn+1 to the root vertex v. As mentioned above T(K1,n)
contains a clique of order n + 1, so by proper colouring procedure it requires minimum of n+1
89
colours, which produces a b-chromatic colouring. Next if we assign the colour cn+1 to the vertex
vi for i = 1, 2, …… n, it will not produce a b-colouring because none of the vertices vi′s are
mutually adjacent to each other. So assigning any new colour to the vertex vi is not possible.
Thus by the colouring procedure the above said colouring produces the maximum and
b-chromatic colouring.
( )1, 1nT k nϕ ∴ = + , n ≥ 3
Example
v
c4
v1
v1'
c1
v2c3
c2
v2'
v3c2
v3'c3
v4'c4
c1v4
c5
Figure 6: ϕ [T(K1,4)] = 5
6.4.2 Structural Properties of Total Graph of Star Graph
• Number of vertices in T(K1,n) = 2n+1
• Number of edges in T(K1,n) = �(���)
�
• Maximum degree of T(K1,n) is ∆ = 2n
• Minimum degree of T(K1,n) is δ = 2
• The number of vertices having maximum degree ∆ in T(K1,n) is n(p∆)=1.
• The number of vertices having minimum degree δ in T(K1,n) is n(pδ)= n
• n vertices with degree n+1.
90
6.5 b-Chromatic Number of Total Graph of Double Star Graph
6.5.1 Theorem
For any integer n≥ 2, the b-Chromatic number of Total graph of Double Star graph is n+1
i.e. φ[T(K1,n,n)] = n+1.
Proof
Consider the Double Star graph K1,n,n i.e. V(K1,n,n) = {v}∪ {vi / 1≤ i≤ n}∪{vi′ / 1≤ i≤ n}.
In T(K1,n,n) by the definition of Total graph, each edge vvi is subdivided by the new vertex ui and
the edge vivi′ is subdivided by the another new vertex ui′ for i=1,2,3..n.
i.e. V [T(K1,n,n)]= {v}∪{vi / 1≤ i≤ n}∪{vi′ / 1≤ i≤ n}∪{ui / 1≤ i≤ n}∪{ui′ / 1≤ i≤ n}
We see that the vertices ui (1≤ i ≤n) are mutually adjacent with each other and the vertices
ui′ are adjacent with vi,vi′ and ui for i=1,2,3..n. Also the vertices u1,u2,u3..un along with root
vertex v induces a clique of order n+1 in T(K1,n,n). Therefore we say that φ[T (K1,n,n)] ≥ n+1.
Now we will prove the other side φ[T(K1,n,n)] ≤ n+1, for this consider a proper colouring of
T(K1,n,n) as follows.
Consider the colour class C={c1,c2,c3…cn,cn+1}. Assign the colour ci to the vertex ui for
i=1,2,3….n and cn+1 to the root vertex v. Here the vertices v,u1,u2,u3…….un realizes its own
colour class, which produces a b-chromatic colouring.
Next, if we assign the colour cn+2 to the vertices vi′ for i=1,2,3..n, the vertices
v,u1,u2,u3..un realizes its own colour class but the vertex vi′ (1≤ i ≤ n) does not realize its own
colour class due to the above mentioned non adjacency condition. So we should assign only the
existing colours to the remaining vertices i.e. assign the colour ci+1 to the vertices vi′ and vi for
i=1,2,3…n-1 and assign the colour c1 to the vertices vn and vn′ then for i=1,2,3..n , ui′
are
assigned with the colour cn. Thus there is no possibility of assigning more than n+1 colours to
every T(K1,n,n) i.e. φ[T (K1,n,n)] ≤ n+1. Therefore we have φ[T (K1,n,n)] = n+1.
Thus by the colouring procedure the above said colouring is maximum and b-chromatic.
91
Example
Figure 7: φ[T(K1,4,4)] = 5
6.5.2 Structural Properties of Total Graph of Double Star Graph
• Number of vertices in T(K1,n,n) = 4n+1
• Number of edges in T(K1,n,n) =�(��)
�
• Maximum degree of T(K1,n,n) is ∆= 2n
• Minimum degree of T(K1,n,n) is δ = 2
• The number of vertices having maximum degree ∆ in T(K1,n,n) is n(p∆)=1
• The number of vertices having minimum degree δ in T(K1,n,n) is n(pδ)= n
6.5.3 Theorem
For any n>2, q[T(K1,n,n)]= �(��)
�
Proof
q[T(K1,n,n)]= Number of edges in Kn + Number of edges not in Kn
= q(Kn)+ Number of edges not in Kn
= ����+7n
= �(� )
� +7n
92
= �� ����
�
= ����
�
= �(��)
�
Therefore q[T(K1,n,n)] = �(��)
�
6.6 b-Chromatic Number of Total Graph of Fan Graph
6.6.1 Theorem
For any integer n>2, the b-Chromatic number of Total graph of Fan graph is n+1.
i.e. φ[T(F1,n)] = n+1.
Proof
Let (X,Y) be a bipartition of F1,n with |x| = 1 and |y| = n. Let X={v} and Y={u1,u2,…un}.
In T(F1,n) by the definition of Total graph each edge vui for 1≤ i ≤ n of F1,n is subdivided by new
vertex vi′ and for i=1,2,3..n-1, ui,ui+1 is subdivided by vertex wi where T={wi / 1≤ i≤ n-1}.
Clearly T is an independent set. The vertex set of Total graph of F1,n is defined as
V [T(F1,n)] = {v}∪{vi ′ / 1≤ i ≤ n}∪{wi / 1≤ i ≤ n-1}∪{ui / 1≤ i ≤ n}
Here the vertices v,v1′,v2′,v3′…vn′ induces a clique of order n+1 in T(F1,n). So that we say
the b-chromatic number of Total graph of Fan graph will have more than or equal to n+1 colours
i.e. φ[T(F1,n)] ≥ n+1. Now we will prove for φ[T(F1,n)] ≤ n+1, for this assign a proper colouring
to these vertices as follows. Consider the colour class C={c1,c2,c3…cn,cn+1}. Assign the colour ci
to the vertex vi′ for i=1,2,3,….n and cn+1 to the vertex v. Here the vertices v,vi′ for i=1,2,3..n
realizes its own colour, which produces a b-chromatic colouring.
Suppose if we assign any new colour to the vertices ui for i=1,2..….n and wi for
i=1,2,3..n-1, the vertices ui and wi does not realizes the new colour due to the non-adjacency
condition. Note that the rearrangement of colours also fails to accommodate any new colour
93
class. Thus there is a possibility of assigning not more than n+1 colours to every total graph of
Fan graph i.e. φ[T(F1,n)] ≤ n+1. Therefore we have φ [T(F1,n)] = n+1.
Thus by the colouring procedure the above said colouring produces a maximal and
b-chromatic colouring.
Example
Figure 8: φ [ T(F1,4)] = 5
6.6.2 Structural Properties of Total Graph of Fan Graph
• Number of vertices in T(F1,n) = 3n
• Number of edges in T(F1,n)= ����� �
�
• Maximum degree of T(F1,n) is ∆= 2n
• Minimum degree of T(F1,n) is δ = 4
6.6.3 Theorem
For any integer n>2, q[ T(F1,n)]= ����� �
�
Proof
q[ T(F1,n)] = Number of edges in Kn+1 + Number of outer edges in T(F1,n) + Number of inner
edges in T(F1,n) + Remaining edges
94
=��(��)� �+2n-1+5n-4+n-2
= n2+n+4n-2+10n-8+2n-4
= ����� �
�
Therefore q[ T(F1,n)]= ����� �
�
6.7 b-Chromatic Number of Total Graph of Bistar
6.7.1 Theorem
The b-Chromatic number of Total graph of Bistar has n +3 colours for every n≥2.
Proof
Consider the Bistar Bn,n. By definition of Bistar, let u1,u2,….un be the n pendant edges
attached to the vertex u and v1,v2,….vn be another n pendant edges attached to the vertex v.
Consider the Total graph of Bistar, by the definition of Total graph each edge uui and vvi is
subdivided by the newly introduced vertices ui′ and vi′ for i=1,2,3…n. Let S be the newly
introduced vertex in between the vertices u and v. In T(Bn,n), both vertex set and edge set of Bn,n