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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 .
Integrated Intelligent Research (IIR)
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
Integrated Intelligent Research (IIR)
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 .
Integrated Intelligent Research (IIR)
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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