Acyclic Colouring of Star Graph Families

Abstract

In this paper, we discuss the acyclic vertex colouring and acyclicchromatic number of middle graph, central graph and total graphof star graph.

Full text

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International Journal of Computer Applications (0975 – 8887)
Volume 7– No.2, September 2010
31

Acyclic Coloring of Star Graph Families
Dr. K. Thilagavathi
Associate Professor
Department of mathematics
Kongunadu Arts and Science College Coimbatore
Tamilnadu-641029, India.
Mr. P. Shanas Babu
Research Scholar
Department of mathematics
Kongunadu Arts and Science College Coimbatore
Tamilnadu-641029, India.


ABSTRACT
In this paper, we discuss the acyclic vertex colouring and acyclic
chromatic number of middle graph, central graph and total graph
of star graph.
General Terms
Middle graph, central graph and total graph of star graph are
denoted by
1,
()
n
M K
,
1,
()
n
C K
and
1,
()
n
T K
respectively.
Keywords
Middle graph, central graph, total graph, acyclic colouring,
acyclic chromatic number.
1. INTRODUCTION
Let G be a graph with vertex set
middle graph [6,9] ofG , denoted by
follows. The vertex set of
, x y in the vertex set of
one of following holds:
( )
V G and edge set ( )
( )
M G is defined as
M G is ( )
V GE G

( )
M G are adjacent in
E G . The
( ) ( )
.Two vertices
( )
M G in case
(i)
in ( )
V G , y is in ( )
, x y Are in
( )
E G and
E G , and , x y are incident inG .
, x y are adjacent inG . (ii) x Is
Let G be a finite undirected graph with no loops and multiple
edges. The central graph [16] of a graphG , ( )
subdividing each edge of G exactly once and joining all the non-
adjacent vertices ofG . By the definition
C G is obtained by
c G
ppq . For any
, p q graph there exist exactly pvertices of degree
vertices of degree 2 in ( )
C G .
1
p
and q
Let G be a graph with vertex set
The total graph [6,9]ofG , denoted by
follows. The vertex set of
, x y in the vertex set of ( )
of the following holds:
( )
V G and edge set ( )
( )
T G is defined as
( )( )
V GE G

.Two vertices
T G are adjacent in ( )
E G .
( )
T G is
T G in case one
(i) , x y are in
in
( )
E G and , x y are adjacent in G . (ii) x is in ( )
( )
E G , and , x y are incident in G .
( )
V G and x is adjacent to y in G .(ii) , x y are
V G , y is in
A proper vertex colouring of a graph is acyclic if every cycle
uses at least three colours [13].The acyclic chromatic number
ofG , denoted bya G , is the minimum k such that G admits
an acyclic k -colouring.
2. ACYCLIC COLOURING OF
1,
()
n
M K

2.1 Theorem
For any star graph
1 n
K, the acyclic chromatic number,
1
1
n
a M K
[
n
,
( )]
.
Proof
Consider the star graph
1,n
K
with
1,n
V K
1, 2, 3,
v v v
, by the definition
,
...n
v v with
v as the root vertex. In middle graph
each edge
i
vv for 1 i
1,
()
n
M K
n of
1 n
K, is subdivided by the vertex
ie in
order
1,
in
()
n
M K
. and the vertices
1, 2, 3,
e e e
,
...n
e v induce a clique of
1
n
(
1,
()
n
M K
i.e.,
1,
( )) { /1} { /1
n

} { }
n

nii
V M K
proper colouring to these vertices as follows. Consider a colour
class
1231
, , ,....n
Cc c cc
. Assign the colour
1,2,..
in and the colour
for
1,2,..
in. The colouring is minimum, as
clique of order
1
n
, minimum
proper colouring. Next we have to prove that the above said
coloring is acyclic. It is obvious that to form a bichromatic cycle
both colours should occur at least twice. But in the above said
colouring, the only colour class which occur atleast twice is
and all other colour classes occur only once in the colouring
procedure. Hence the above said colouring is acyclic.
vieiv
. Now assign a
ic to the vertex
ie for
1
nc
to the vertex v and to
(
M K
iv
1,
)
n
contains a
1
n
colours are required for its
1
nc

Thus
1
1
n
a M K
[
n
,
( )]
.
Example
Figure 1

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International Journal of Computer Applications (0975 – 8887)
Volume 7– No.2, September 2010
32

1
1
n
a M K
[
n
,
( )]

3. ACYCLIC COLOURING OF
1,
()
n
C K

3.1 Theorem
For the graph
1,n
K
the acyclic chromatic number
1,
,3
n
a C Kn n
.
Proof
Consider the star graph
1,n
K
with
1,n
V K
1,2, 3, , 0
...n
u uu u u
with
0 u as the root vertex. In
1,
()
n
C K
,
let
,0
iu
represents the newly introduced vertex in the edge
connecting
colouring to these vertices as follows. Consider a colour
class
123
, , ,....n
C c c cc
. Assign the colour
1,2...
in and the colour
colouring excludes the newly introduced vertices. The colour
2c is assigned to the newly introduced vertices
and the colour
u
iu and
0 u for
1,2...
in. Now assign a proper
ic to the vertex
iu for
1c to the vertex
0 u . Such a
,0
iu for
2
i

3c is assigned to
contains the subgraph
required for its proper colouring. Next we have to prove that the
above said coloring is acyclic. It is obvious that to form a
bichromatic cycle both colours should occur at least twice. So in
the above said colouring, the color classes
induce a 2-chromatic cycle (it occur only once in the colouring
procedure). Now we need to examine the subgraphs induced by
,
ij
c c
for
1,2
i
and
2,3
j
withi
2-chromatic cycle or not.
2,0
. The colouring is minimum,
as
1,
()
n
C K
n
K , minimum n colours are
,4
kckn never
j whether they induce a
Case 1. If
is a tree with
1
n
i
and
2
j
, then the sub graph induced by
pendent vertices, which is clearly a forest.
12
,
c c

3
Case 2. If
is the disjoint union of
forest.
Case 3. If
,
c c
is the union of a tree with
also form a forest.
Thus any pair of the colour class will never induce a 2-chromatic
cycle in the graph.
1
i
and
3
j
, then the subgraph induced by
13
,
c c

2P with itself. That is it forms a linear
2
j
and
3
j
, then the subgraph induced by
3
n
isolated vertices, which
23
Therefore,
1,
,3
n
a C K n n
.

Note:
1,
3,
n
a C K
for
2
n
.
Example




Figure 2
1,4
4
a C K

4. ACYCLIC COLOURING OF
1,
()
n
T K

4.1 Theorem
For any star graph
1 n
K, the acyclic chromatic number,
1
1
n
a T K
[ (
n
,
)]
.
Proof
Conside the star graph
1,n
K
with
1,n
(
V K
1, 2, 3,
v v v
, by the definition
,
...n
v v with
v as the root vertex. . In Total graph
1,
)
n
T K
each edge
i
vv for 1 in of
1,n
K
is subdivided by the vertex
ie in
order
1,
()
n
T K
and the vertices
1, 2, 3,
e e e
,
...n
e v induce a clique of
()
n
T K
.
} { }
nv

. Now assign a
1
n
in
1,
i.e.,
1,
(()) { /1} { /1
n

nii
V M K
proper colouring to these vertices as follows. Consider a colour
class
1231
, , ,....n
Cc c cc
. Assign the colour
1,2,..
in and the colour
assign the colour
viei
ic to the vertex
ie for
1
nc
to the vertex v . Now
n and
(
T K
1c to
iv for 2 i
2c to the remaining
)
n
contains a
colours are required for its
vertex
clique of order
proper colouring. In the above said colouring, the color classes
,31
kn
never induce a 2-chromatic cycle (it occur only
once in the colouring procedure). Also the subgraphs induced by
,
c c
is the union of a path
which is a forest. Thus any pair of the colour class will never
induce a 2-chromatic cycle in the graph. That is the colouring is
acyclic.
1v . The colouring is minimum, as
1
n
, minimum
1,
1
n
kc
12
3P with
2
n
isolated vertices,

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International Journal of Computer Applications (0975 – 8887)
Volume 7– No.2, September 2010
33

Thus
1
1
n
a T K
[ (
n
,
)]
.
Example



Figure 3
1
1
n
a T K
[ (
n
,
)]
.
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