b-Chromatic Number of Subdivision Edge and Vertex Corona

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International Journal of Computing Algorithm Volume: 03, Issue: 01 June 2014, Pages: 44-47 ISSN: 2278-2397

b-Chromatic Number of Subdivision Edge and Vertex Corona T.Pathinathan1, A.Arokia Mary2, D.Bhuvaneswari2 1

Department of Mathematics,Loyola college,Chennai Department of Mathematics, St.Joseph’sCollege, Cuddalore, India Email:arokia68@gmail.com,bhuvi64@gmail.com

2

Abstract - In this paper, we find that the b-chromatic number on corona graph of subdivision-vertex path with path. Then corona graph of any graph with path, cycle and complete graph and cycle with path.

Keywords: b-chromatic number, corona graph, subdivisionedge corona, subdivision-vertex corona, edge corona.

I. INTRODUCTION The b-chromatic number of a graph G , denoted by

b (G) , is

DEFINITION2.2. Let

the maximal integer k such that G may have a b-coloring with k colors. This parameter has been defined by Irving and Manlove. The subdivision graph S (G) of a graph

G is

sets of

n1 and n2

graph obtained by inserting a new vertex into every edge of G ,

and

vertices,

The edge corona

the

G1

G2

m1 of

be two graphs on disjoint

and

G2

G1

is defined as the and

G2

be

two

graphs.

th

Let

Example: Let

G1

complete graph

k2

corona

G1 G2 is

the graph obtained by joining each th

every vertex of the i copy of

Vi to

G2 , 1  i  k .

44

G1

to

G2 .

be the cycle of order 4 and

G2

of order 2. The two edge coronas

G2

V(G1 )  v1 , v2 ,..., vk  and take k copies of G2 . The

copies of

th

II. PRELIMINARIES DEFINITION2.1.Let G1 and

m1

and then joining two end-vertices of the i edge of

every vertex in the i copy of

In two new graph operations based on subdivision graphs, subdivision-vertex join and subdivision-edge join. The corona of two graphs was first introduced by R.Frucht and F.Harary in [11]. And another variant of the corona operation, the neighbourhood corona, was introduced in [12].

edges respectively.

G1 and G2

graph obtained by taking one copy of

we denote the set of such new vertices by I (G) .

m2

G1

be the .

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International Journal of Computing Algorithm Volume: 03, Issue: 01 June 2014, Pages: 44-47 ISSN: 2278-2397

III. SUBDIVISION-VERTEX AND SUBDIVISIONEDGE CORONA

1

2

DEFINITION 3.1. The Subdivision-vertex Corona of two

G2 , denoted by

vertex-disjoint graphs

G1

graph obtained from

S (G1 ) and V(G1 ) copies

and

th

th

G2 ,

of

V(G1 )

vertex-disjoint by joining the i vertex of vertex in the i copy of

is the

2

1

all

3

to every

G2 .

2

3

1

DEFINITION3.2.The Subdivision-edge Corona of two vertex disjoint graphs

G1

obtained from

S (G1 )

and

th

G1 is a

graph on

n2

graph on

th

vertex of

is the graph

I(G1 )

n1

IV. VERTEX CORONA

G2 , all vertex-

to every vertex in

m1 edges

vertices and

m2 has

2m1  n1 (n2  m2 )

4.1 GRAPHS WITH PATH

and

b (G Pn )  

G2 is a

n1 (1  n2 )  m1 vertices

and

Let V(G

m1 (2  n2  m2 )

vertices and

for n  3 for n  3

Pn )  vi :1  i  n  Uij :1  i  n;1  j  n .

{

be a path of

vertices and

vi  V(G)

be a path

u

ij

{ {

Pn .

copy of

i.e) every vertex

is adjacent to every vertex from the set

:1  j  n .

.

Assign the following

PROOF. Let ( ) ( )

n

to every vertex of

)

n vertices. Then

By the definition of Corona graph each vertex of G is adjacent

edges.

(

n 1, n

PROOF: Let V(G)  v1 , v2 ,..., vn  and V(Pn )  u1 , u2 ,..., un  .

edges then the subdivision-

edges, and the Subdivision-edge Corona

m1 (1  n2 )  n1

THEOREM 3.3. Let of vertices. Then

2

)

THEOREM4.1.1 Let Gbe a simple graph on

vertices and

vertex Corona

has

denoted by

(

G2 .

the i copy of Let

G2

and I(G1 ) copies of

disjoint, by joining the i

5

4

} and ( } {

)

{

} let

} By the definition of corona graph, each vertex of adjacent to every vertex of a copy of

is

Assign the following n-coloring for  For , assign the color  For , assign the color  For , assign the color  For , assign the color  For assign the color  For , assign to vertex

to . to , to to , to , one of allowed colors.

Define (  For  For  For  For

to . to to to

) , assign the color , assign the color , assign the color , assign the color

as b-chromatic.

n -coloring for G Pn

as b-chromatic



For 1  i  n , assign the color

ci to vi .



For 1  i  n , assign the color



For 1  i  n , assign the color

ci to u1i , i  1. . ci to u2i , i  2. .



For 1  i  n , assign the color

vi to uni , i  n. .



For 1  i  n , assign the vertex colors-such color exists because

uii one of allowed 2  deg(u ii )  3 and

n 3. b (G Pn )

Let us assume that

b (G Pn )  n  1 n  3 vertices of degree

n

in

is greater than

there must be at least

G Pn

n ie)

n 1

all with distinct colors, and

each adjacent to vertices of all of the other colors. But then

these must be the vertices v1 , v2 ,..., vn  . Since these are only

45

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ones with degree at least

International Journal of Computing Algorithm Volume: 03, Issue: 01 June 2014, Pages: 44-47 ISSN: 2278-2397

n . This is a contradiction. b-coloring

Assign the following

n  1 colors is impossible.

with Thus

we

have

b (G Pn )  n .

Hence

b (G Pn )  n ,



For 1  i  n , assign the color

ci



For 1  i  n , assign the color

ci

Define b-coloring of G P4 ( V(G) )  4 with 5 colors in the



For 1  i  n , assign to vertex

colors-such color exists, because For 1  i  5 , assign the color

ci

to

vi .



For 1  l  5 , assign the color

cl

to

u1l , l  1.



For 1  l  5 , assign the color

cl

  

b (G Cn )  n .

u2l , l  2.

For 1  l  5 , assign the color cl to u3l , l  3.

For 1  l  5 , assign the color cl to u4l , l  4.

For 1  l  5 , assign the color cl to u5l , l  5.

n,

to

vi .



following way: 

to

as b-chromatic:

u1i , i  1.

For 1  i  n , assign the color ci to u2i , i  2.

For 1  i  n , assign the color ci to uni , i  n.



n  3 .

n -coloring for G Cn

ie)

n 1

uii one of allowed deg(u ii )  3 and n  3

b (G Cn ) is

Assume that

b (G Cn )  n  1 n  3 , in G

n

vertices of degree

to

.

greater than

there must be at least

Cn ,

all with distinct

colors, and each adjacent to vertices of all of the other colors.

But then these must be the vertices v1 , v2 ,..., vn  . Since these are only ones with degree at least

We have

b-coloring with

b (G P4 )  5 . Hence b (G Pn )  n n  3 .

We

2 3 4 5

have

n . This is the contradiction

n  1 color is impossible.

b (G Cn )  n .

Hence

b (G Cn )  n ,

n  3 . (

)

1 5 4

2 5

3 1

(

3 1 2

4

)

4 1

5

3

2

2

G be

Let Let

(

)

1

3 4

5 3 4.3. CYCLE WITH PATH

a simple graph on

THEOREM 4.3.1Let Cn be a cycle of

n vertices

path of

V(G Cn )  vi :1  i  n  uij :1  i  n;1  j  n .

By

.Let

definition of Corona graph, each vertex of G is adjacent to

adjacent to every vertex from the set

u

ij

vertices. Then

b (Cn Pn )  n

Let V(Cn )  v1 , v2 ,..., vn  and V(Pn )  u1 , u2 ,..., un 

the

Cn . i.e., every vertex vi  V(G)

n

n vertices and Pn be a

Proof:

V(G)  v1 , v2 ,..., vn  and V(Cn )  u1 , u2 ,..., un  .

every vertex of a copy of

4

1

1

n  3 . Then b (G Cn )  n .

Proof:

4 2

4.2. GRAPH WITH CYCLE THEOREM4.2.1. Let

2

3

1 2 3 42 2 2

V(Cn Pn )  vi :1  i  n  uij :1  i  n;1  j  n ,

By the

definition of Corona graph, each vertex of G is adjacent to

is

every vertex of a copy of

:1  j  n .

Pn . i.e., every vertex vi  V(Cn )

adjacent to every vertex from the set 46

u

ij

is

:1  j  n .

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b (Cn Pn )  n .

International Journal of Computing Algorithm Volume: 03, Issue: 01 June 2014, Pages: 44-47 ISSN: 2278-2397 

Assign the following b-coloring for



Cn Pn .

For 1  i  n , assign the color



ci

to



For 1  i  n , assign the color

ci

to



For 1  i  n , assign the color

ci

to

vi .

u1i , i  1. u2i , i  2.

 

1  i  n , assign the color ci to uni , i  n.

For 1  i  n , assign to vertex uii one of allowed colors such color exists, because 2  deg(u ii )  3 and n  3 .

1  l  n , assign the color cl to unl , l  n.

For 1  l  n , assign the color cn 1 and ull Therefore,  b G  K n   n  1.

For

Let us assume that

For



 n , assign the color cl to u3l , l  3.

For 1  l  n , assign the color cl to u4l , l  4.

…………………………………… …………………………………… …………………………………… ...………………………………….

………………………………… …………………………………. …………………………………. …………………………………… 

For 1  l

 b G  K n 

 b G  K n   n  2 , of degree

is greater than

there must be at least

n  1 , i.e.,

n  2 vertices

n  1 in G K n , all with distinct colors, and each

adjacent to vertices of all of the other colors. But then these We have

b (Cn Pn )  n . Hence b (Cn Pn )  n .

must be the vertices

We have

b (C3 P2 )  3 .

degree at least

2

v1 , v2 ,...vn , since these are only ones with

n  1 . This is the contradiction, b-coloring with

n  2 colors is impossible.  b G  K n   n  1. . Hence,.

3

Thus,

we

have

1 V. CONCLUSION

2

1

(

)

In this existing subdivision-vertex corona graphs, they used spectrum. Here we study about subdivision- vertex corona graph using b-chromatic number.

1

3 2

3

REFERENCE 4.4. GRAPH WITH COMPLETE GRAPH Theorem4.4.1 :Let G be a simple graph on

 b G  K n   n  1.

[1]

n

vertices. Then

[2] [3]

Proof.

[4]

Let V(G)  v1 , v2 ,..., vn  and V(K n )  u1 , u2 ,..., un  .

the b-colring of cographs and

vi  V(G)

u

ij

Kn .

[5]

i.e., every vertex

is adjacent to every vertex from the set

:1  j  n .

Assign the following

n  1 -coloring

for

G Kn

as b-

chromatic: 

For 1  i  n , assign the color



For 1  l



ci

 n , assign the color cl For 1  l  n , assign the color cl

to

vi .

to

u1l , l  1.

to

u2l , l  2.

P4 -sparse

graphs, Graphs and

Combinatorics, 25(2) (2009), 153-167.

R.W. Irving and D.F. Manlove, The b-Chromatic number of a graph, Discrete Appl.Math. 91 (1999), 127-141.

[6] M. Kouider and A. El Sahili, About b-colouring of regular graphs, Rapport de Recherche No 1432, CNRS-Universite Paris SudLRI, 02/2006.

[7] Marko Jakovacand SandiKlavzar, The b-Chromatic number of cubic graphs, Graphs and Combinatorics, 26(1) (2010) 107-118.

[8] Venkatachalam.M and vernoldVivin.J, The b-Chromatic number of star graph families, Le Mathematiche, 65(1) (2010), 119-125.

[9] D. M. Cvetkovic, M. Doob, H. Simic, An Introduction to the Theory of Graph Spectra, Cambridge University Press, Cambridge, 2010.

[10] G.Indulal, Spectrum of two new joins of graphs and iinfinte families of integral graphs, KragujevacJ.Math. 36 (2012) 133-139.

[11] R. Frucht, F.Haray, On the corona of two graphs, Aequationes Math. 4 (1970) 322-325.

[12] I. Gopalapillai, The spectrum of neighborhood corona of graphs, Kragujevacjournal of Mathematics 35 (2011) 493-500.

} { ) { Let ( }.

By the definition of corona graph, each vertex of G is adjacent to every vertex of a copy of

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