The locating chromatic number of non homogeneous amalgamation of stars

  Far East Journal of Mathematical Sciences (FJMS) © 2014 Pushpa Publishing House, Allahabad, India Published Online: November 2014 Available online at http://pphmj.com/journals/fjms.htm Volume 93, Number 1, 2014, Pages 89-96

  

THE LOCATING-CHROMATIC NUMBER OF

NON-HOMOGENEOUS AMALGAMATION OF STARS

Asmiati

  Department of Mathematics Faculty of Mathematics and Natural Sciences Lampung University Jl. Brojonegoro no. 1, Bandar Lampung, Indonesia e-mail: asmiati308@yahoo.com

  

Abstract

Let G be a connected graph and c be a proper coloring of G. The color

c v

code ( ) of vertex v in G is the ordered k-tuple distance v to every

  Π C i

color classes i for ∈ [ 1 k , ] . If all distinct vertices of G have

distinct color codes, then c is called a locating-coloring of G. The

  G

  ( )

  

locating-chromatic number of graph G, denoted by χ L is the

smallest k such that G has a locating-coloring with k colors. In this

paper, we discuss the locating-chromatic number of non-homogeneous

amalgamation of stars.

  Research topics about the locating-chromatic number of a graph have grown rapidly since its introduction in 2002 by Chartrand et al. [8]. This concept is combined from the graph partition dimension and graph coloring.

  Specially for tree, Chartrand et al. [9] determined the locating-chromatic

  Received: May 17, 2014; Accepted: September 1, 2014 2010 Mathematics Subject Classification: 05C12, 05C15.

Keywords and phrases: color code, locating-chromatic number, non-homogeneous

  90 Asmiati numbers of some particular trees. Asmiati et al. [2, 3] obtained the locating- chromatic number of firecracker graphs and amalgamation of stars, respectively. Next, Welyyanti et al. [10] discussed the locating-chromatic number of complete n-arry tree.

  In general graphs, Asmiati and Baskoro [1] characterized graph containing cycle with locating-chromatic number three. Behtoei and Omoomi [4-6] determined the locating-chromatic number of Kneser graphs, join of the graphs, and Cartesian product of graphs, respectively. Next, Baskoro and Purwasih [7] discussed the locating-chromatic number of corona product of graphs.

  Let G =

  V , E be a connected graph and c be a proper k-coloring ( )

  Π =

  of G using the colors

  1 , 2 , ..., k for some positive integer k. Let

  { C , C , ..., C } be a partition of V ( ) G which is induced by coloring c. The

  1 2 k color code c ( ) v of vertex v in G is the ordered k-tuple ( ( d v , C ) , ...,

  Π

  1 d ( v , C )) , where d ( v , C ) min { d ( v , x ) x C } for i 1 k , . If all distinct

  = | ∈ ∈ [ ] k i i

  vertices of G have distinct color codes, then c is called a locating k-coloring

  G

  of G. The locating-chromatic number, denoted by χ ( ) is the smallest k

  L such that G has a locating k-coloring.

  Let G = K be a star graph for every i ∈ [

  1 , k ] and n ≥ 1 . Non- i 1 , n i i

  homogeneous amalgamation of stars, denoted by S , k ≥

  2 is a k , ( n , n , ..., n )

  1 2 k graph formed by identifying a leaf of G for every i 1 k , . We call the ∈ [ ]

i

identified vertex as the center of S , denoted by x. The vertices

k , ( n , n , ..., n )

  1 2 k of distance 1 from the center as the intermediate vertices, are denoted by , l i for i 1 k , . We denote the jth leaf of the intermediate vertex l by l for ∈ [ ] i ij

  ∈ − = ≥ j [ 1 , n 1 ] . In particular, when n m , m 1 for all i, we denote the j i homogeneous amalgamation of k isomorphic stars K by S .

  1 , m k , m

The locating-chromatic number of homogeneous amalgamation of

stars and its monotonicity property have been studied by Asmiati et al. [3].

  The Locating-chromatic Number of Non-homogeneous …

  91 Motivated by this, in this paper, we determine the locating-chromatic number of non-homogeneous amalgamation of stars. These results are generalization of results in [3].

  

The following basic theorem was proved by Chartrand et al. in [8]. The

set of neighbours of a vertex v in G is denoted by N ( ) v .

  Let c be a locating-coloring in a connected graph G.

  Theorem 1.1.

  If u and v are distinct vertices of G such that d u , w = d v , w for all ( ) ( ) w ∈

V ( ) { G − u , v } , then c ( ) u ≠ c ( ) v . In particular, if u and v are non-

adjacent vertices of G such that N ( ) u N ( ) v , then c ( ) u c ( ) v .

  = ≠

Corollary 1.1. If G is a connected graph containing a vertex adjacent to

k leaves of G, then χ ( ) ≥ + G k 1 .

  L

  2. Main Results

In this section, first we give some basic lemmas about the locating-

chromatic number of non-homogeneous amalgamation of stars. Let n = max max { n , n , ..., n } .

  1 2 k

Lemma 2.1. Let c be a proper n -coloring of S , where

max k , ( n , n , ..., n )

  1 2 k

  ≥ n 1 , i 1 k , .

  k

  2 , ≥ and ∈ [ ] The coloring c is a locating-coloring if and

  i only if c ( ) l = c ( ) l , i ≠ implies j { c ( ) l | s = 1 , 2 , ..., n − 1 } and { ( ) c l |s =

i j is i js

  1 , 2 , ..., n − 1 are distinct.

  } j

  = | =

  Proof. Let P = { c ( ) l | s = 1 , 2 , ..., n − 1 } and Q { ( ) c l s

  1 , 2 , ...,

  is i js n −

  1 } . Let c be a proper n -coloring of S where ≥

  k

  2 ,

  max k , ( n , n , ..., n ), j

  1 2 k

  ≥ = ≠ =

  n

  1 , and c ( ) l c ( ) l for some i n . Suppose that P Q . Because

  i i j d ( l , u ) = d ( l , u ) for every u ∈

V \ {{ l | s =

  1 , 2 , ..., n − 1 } { l | s = 1 ,

  i j is i ∪ js

  − 2 , ..., n 1 }} , then the color codes of l and l will be the same. So c is not

  j i j a locating-coloring, a contradiction. Therefore, P ≠ Q .

• Clearly,
• If

  ≠ From all the above cases, we see that the color code for each vertex in

  2 1 k n n n k

  Next, we will determine locating-chromatic number of non- homogeneous amalgamation of stars ( ). ..., , , ,

  1 is unique, thus c is a locating-coloring.

  2

  S ..., , , ,

  k n n n k

  ( )

  Π Π

  ( )

  ( ) ( ) . is l c x c

  contains exactly one component of value 1. Thus,

  Π

  is l c

  contains at least two components of value 1, whereas ( )

  Π

  ( ) x c

  is = l c x c then color code of

  S Recall definition of

  , ..., , , ,

  ( ) ( )

  − = max

  ≠

  max ,

  amalgamation of stars . ,

  

j

G induce subgraph of homogeneous

  for some t (value t may be 0). Therefore,

  1

  ,

  j n j j K t G

  2 1 k n n n k

  = where

  = − =

  ,

  1 max

  

1

,

  1

  G K i

  S ∪ ∪ k i n j j n

  ,

  2 r th-ordinate.

  − S t j n t

  1 ∈ are unique.

  ≠

  = for some . l k l l

  ( ) ( ) ls kt l c l c

  xth-ordinate and yth-ordinate.

  ≠ because their color codes differ in the

  Π Π

  ( ) ( ) j i l c l c

  2

  Case 1. If ( ) ( ) , l k l c l c

  S V v ..., , , ,

  ) k n n n k

  such that ( ) Q x P x ∉ ∈ , or ( ) . , Q y P y ∉ ∈ We will show that color codes for every ( ( )

  j i = l c l c . j i ≠ Since , Q P ≠ there are color x and color y

  Consider ( ) ( ) ,

  max ≥ n Π

  92 Let Π be a partition of ( ) G V into color classes with .

  Asmiati

  We divide into two cases:

  =

  and

  2

  1 r th-ordinate

  because their color codes are different at least in the

  Π

  ( ) ls l c

  Π kt l c

  ≠

  ≠ Then ( )

  1 r r

  = with .

  then by the premise of this theorem, . Q P ≠ So ( ) ( ) .

  2 r l c l

  ( )

  = and

  1 r l c k

  Case 2. Let ( )

  ≠

  Π Π

  ls kt l c l c

• If

  The Locating-chromatic Number of Non-homogeneous …

  93

  ( ) + If c is a locating n a -coloring of S Lemma 2.2. max k , ( n , n , ..., n )

  1 2 k n

• max

  −

  n a

  1  

  max

  ≥ = − ≥ +

  for k

  2 , n ≥

  1 , and I ( ) a ( n a

  1 )   , a ,

  i max ∑ n − i

   max 

  i =

  1 a ∈ Z , then k ≤ I ( ) a .

• Proof. Let c be a locating ( n a )
• coloring of S By max k , ( n , n , ..., n ).

  1 2 k

  Lemma 2.1, maximum value t for subgraph of homogeneous amalgamation

• n a −

  1  max 

• of stars s for some j, is n a − 1 . Because

  ( )   t , n − j max max

  − −

  n j

  1  

  max

  = − = j +

  j ,

  1 , 2 , ..., n 1 and i 1 , then the maximum number of k is

  max

  max n a −

• n

  1  

  max ( + n a −

  1 ) =

  I ( ) a . As a result, k I ( ) a .

    ≤

  max ∑ n − i

   

  max i

  1 =

  Before we discuss about the locating-chromatic number of non- homogeneous amalgamation of stars S , we recall to construct

  k , n , n , ..., n

  ( )

  1 2 k

  the upper bound of the locating-chromatic number of homogeneous amalgamation of stars was determined by Asmiati et al. [3].

  χ = ≤ ≤ − ≥

  The first result. We got that ( S ) m for

  2 k m 1 , m 3 .

  L k , m

  To construct the upper bound of χ ( S ) for 2 ≤ k ≤ m − 1 , we can

  

L k , m

  = = i = + assign c ( ) x 1 and c ( )

  l

  1 for i 1 , 2 , ..., k . To make sure that the

  i

  leaves will have distinct color code, we assign { l | j = 1 , 2 , ..., m − 1 } by

  ij {

• 1 ,

  2 , ..., m } { \ i 1 } for any i. By Lemma 2.1, these colors are a locating- coloring.

  − + m a

  1  

• The second result. Let H ( ) ( a = m a − 1 ) , m

  2 . Then

    ≥ −

  m

  1  

  χ = − < ≤

• 1 ) k H ( )

  ( S ) m a for H ( a

  a , a ≥ 1 . To determine the L k , m

  upper bound of χ ( S ) for H ( a

  1 ) k H ( ) a , a ≥

  1 . We assign

  − < ≤ L k , m

  = +

  c ( ) x

  1 ; c ( )

  l by

  2 , 3 , ..., m a in such a way that the number of

  94 Asmiati the intermediate vertices receiving the same color t does not exceed

  m a − +

  1  

  . If c l c l , i ≠ n , then c l | j = 1 , 2 , ..., m − 1 ≠   ( ) = ( ) { ( ) }

  i n ij

  −

  m

  1  

  | = −

  { ( ) c l j

  1 , 2 , ..., m 1 } . So, by Lemma 2.1, c is a locating-coloring.

  nj

  Now, we will discuss about the locating-chromatic number of non- homogeneous of amalgamation of stars.

  max n a −

• n

  1  max 

  = − a ≥ , + Let

  Theorem 2.3. n a I ( ) a (

  1 )   ,

  max ∑

  −

  n i

   

  max i =

  1

  ∈

  

a Z . The locating-chromatic number of non-homogeneous amalgamation

  ≥ ≥ + ≤ −

  of stars S where k

  2 , n 1 , and A n a 1 is

  k , ( n , n , ..., n ), i max

  1 2 k n for

  2 ≤ k ≤

  I ( ) ;

  

  max

  χ ( S ) =

  L k , ( n , n , ..., n ) 

  2 k

• 1

  − < ≤ ≥

  n a for I ( a

  1 ) k

  I ( ) a , a

  1 ; 

  max

  

max L k , ( n , n , ..., n )

  1 , then χ ( S ) = A + + whereas, if A > n a − 1 .

  1 2 k Proof. First, we determine the trivial lower bound for 2 k I ( ) .

  ≤ ≤

  Because each vertex l is adjacent to ( n − 1 ) leaves for i ∈ 1 k , , then

  [ ] i max

  χ ≥ by Corollary 1.1, ( S ) n .

  L k , ( n , n , ..., n ) max

  1 2 k

  ≤ Next, we determine the upper bound of S for

  2

  k , ( n , n , ..., n )

  1 2 k

  ≤

  k I ( ) . Let c be a n -coloring of S Because non- max k , ( n , n , ..., n ).

  1 2 k

  homogeneous amalgamation of stars S contains subgraph of

  k , ( n , n , ..., n )

  1 2 k

  homogeneous amalgamation of stars S for some t ≠ , then color

  t , n j − max

  vertices of S for every j , n

  − max

  2 follow construction the ∈ [ − ] t , n j max

  upper bound of the first results in [3]. Next, for j = n −

  1 , color the max

  − intermediate vertices by 2 , 3 , ..., n 1 , respectively. Therefore, c is a

  max

  χ ≤ ≤ ≤ locating-coloring. So, ( S ) n for 2 k I ( ) .

  L k , ( n , n , ..., n ) max

  1 2 k

  The Locating-chromatic Number of Non-homogeneous …

  95 We improve the upper bound for

  

I ( a −

  1 ) ≤ k ≤

  I ( ) a , a 1 . Because ≥

  I ( a 1 ) , then by Lemma 2.2, ( S ) n a . On > − χ ≥

• k

  L k , ( n , n , ..., n ) max

  1 2 k

  the other hand, if k

  I ( ) a , then by Lemma 2.2, ( S ) > χ ≥

  L k , ( n , n , ..., n )

  1 2 k

• a + n 1 . As a result, χ ( S ) ≥ n a , if

  I ( a − 1 ) ≤ + k max L k , ( n , n , ..., n ) max

  1

2 k

  ≤ ≥

  I ( ) a , a 1 .

  − Next, we shall show the upper bound of S for

  I ( a

  1 )

  k , ( n , n , ..., n )

  1 2 k

  ≥

  ≤ k ≤ I ( ) a , a

  1 . The vertices S for every j ∈ [ , n −

  2 ] , we t , n − j max max

  give color follow construction the upper bound of the second result in [3] using ( n a ) = n + colors. Otherwise, for j − 1 , the vertices l are

  

max max i

• a − χ ≤ colored by

  2 , 3 , ..., n 1 , respectively. Thus, ( S )

  max L k , ( n , n , ..., n )

  1 2 k

  − ≤ ≤ ≥ +

  n a for I ( a

  1 ) k

  I ( ) a , a 1 . max

  1 2 k

  1 . By Corollary 1.1, we obtain χ S

( ) + Let A > n a −

max L k , ( n , n , ..., n )

• 1 , since x is adjacent to A leaves. To determine the upper ≥ A

  2 , 3 , ..., A + bound, the vertices in A are colored by 1 , respectively, and the colors for the other vertices alike two cases before it. So,

  χ ( + S ) ≤ A 1 .

  L k , ( n , n , ..., n )

  1 2 k References

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