ON THE LOCATING-CHROMATIC NUMBERS OF NON-HOMOGENEOUS CATERPILLARS AND FIRECRACKER GRAPHS

  Far East Journal of Mathematical Sciences (FJMS) © 2016 Pushpa Publishing House, Allahabad, India Published Online: September 2016 http://dx.doi.org/10.17654/MS100081305 Volume 100, Number 8, 2016, Pages 1305-1316

ISSN: 0972-0871

  

ON THE LOCATING-CHROMATIC NUMBERS OF

NON-HOMOGENEOUS CATERPILLARS

AND FIRECRACKER GRAPHS

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

  

Abstract

We determine the locating-chromatic numbers of non-homogeneous

caterpillars and firecracker graphs.

1. Introduction

  The notion of locating-chromatic number of a graph was introduced by

Chartrand et al. [8]. Let G be a finite, simple, and connected graph. Let c be a

C C C

  V G

  Π =

  

proper k-coloring of G and { , , ..., k } be a partition of ( )

  

1

  2 V G C

  induced by c on ( ) , where i is the set of vertices receiving color i. The

  color code c v d v C d v C d v C ( ) of v is the ordered k-tuple ( ( , ) ( , , ) , ..., ( , k ) ) ,

  Π

  1

  2 d v C d v x x C

  where ( , ) = min { ( , ) | ∈ } for any i. If all distinct vertices of G _{i i}

  Received: May 13, 2016; Accepted: June 29, 2016 2010 Mathematics Subject Classification: 05C12, 05C15. Keywords and phrases: locating-chromatic number, caterpillar, firecracker graph.

  1306 Asmiati have distinct color codes, then c is called a locating-chromatic k-coloring of

  

G (k-locating coloring, in short). The locating-chromatic number, χ ( ) G is

L the smallest k such that G has a locating k-coloring.

  Chartrand et al. [8] determined the locating-chromatic number for paths, cycles, complete multipartite graphs and double stars. Behtoei and Omoomi [7] discussed the locating-chromatic number for Kneser Graph. Specially for amalgamation of stars, Asmiati et al. [1, 4] determined locating-chromatic number for homogeneous amalgamation of stars and non-homogeneous amalgamation of stars, respectively. Furthermore, the locating-chromatic number of the operation of two graphs is discussed by Baskoro and Purwasih [6]. They determined the locating-chromatic number for a corona product of two graphs.

  Chartrand et al. [8] characterized graphs have locating-chromatic number −

  n

  1 . They also determined graphs whose locating-chromatic numbers are bounded by n − 2 . Moreover, Asmiati and Baskoro [3] characterized all maximal graphs containing cycle. In general, characterization of all tress with locating-chromatic number 3 is given in Baskoro and Asmiati [5].

  Asmiati et al. [2] determined the locating-chromatic number for homogeneous firecracker graphs, Motivated by these results, we determine the locating-chromatic numbers of non-homogeneous caterpillars and firecracker graphs.

  The following results were proved by Chartrand et al. in [8]. We denote the set of neighbors of a vertex v in G by N ( )

  V .

  

Theorem 1 [8] . Let c be a locating-coloring in a connected graph G

  =

  (

V , E ) . 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 [8] . If G is a connected graph containing a vertex adjacent to m end -vertices of G, then χ ( ) G ≥ + m 1 .

  L

  On the Locating-chromatic Numbers of Non-homogeneous … 1307 Corollary 1 gives a lower bound for the locating-chromatic numbers of a general graph G.

  

2. Locating-chromatic Number of Non-homogeneous Caterpillar

  In this section, we discuss about the locating-chromatic number of non- homogeneous caterpillar. Let P be a path with

  V ( ) { P = x , x , ..., x } m m

  1 2 m

  = and E ( ) { P x x , x x , ..., x x } . A non-homogeneous caterpillar is

  m

  1

  2

  2 3 m − 1 m =

  obtained by connecting n pendant vertices ( a , j

  1 , 2 , ..., n ) to one i ij i

  ≤ ≤ particular vertex x of path P , where 1 i m , which we denote by i m

  

C ( m ; n , n , ..., n ) . The non-homogeneous caterpillar C ( m ; n , n , ..., n )

  1 2 m

  1 2 m consists of vertex set V ( C ( m ; n , n , ..., n ) ) { = x | 1 ≤ i ≤ m } { a | 1 ≤ i

  1 2 m i ∪ ij

  , 1 ; , , ...,

  1 ≤ m ≤ j ≤ n } and edge set E ( C ( m n n n ) ) { = x x | ≤ i ≤ m

  1 | ≤ ≤ ≤ ≤ 1 } { x a

  1 2 m i i

+ i

  1 i m , 1 j n } . − ∪ i ij i

  Let K , with the vertex x as the center, be a subgraph of 1 , n i i

  C ( m ; n , n , ..., n ) . We denote the set of vertices and edges by V ( K ) =

  1 2 m 1 , n i

  |1 ≤ ≤ { a j n } { } x and E ( K ) { = x a |

  1 ≤ j ≤ n } , respectively. Thus,

  ij i ∪ i 1 , n i ij i i

  

C ( m ; n , n , ..., n ) contains m stars K with x as a center. If n =

  1 2 m 1 , n i max i

  max { n , n , ..., n } , then subgraph K is called the maximum star

  1 2 m 1 n , max

subgraph in the non-homogeneous caterpillar C ( m ; n , n , ..., n ) . If there

  1 2 m

are p subgraphs K , then every subgraph, from left to right, are denoted

  1 n , max i

  ≤ ≤ by K , where 1 i p .

  1 , n maks

  Definition 1. Let , ⊂ ; , , ..., , where 1 ≤ ≠ K K C ( m n n n ) i j

  1 , n 1 , n

  1 2 m i j

  ≤ = ≠ m . If n n n , such that i j max

  (1) d ( x , x ) = d ( x , x ) , with x is the center of K , or i m j m m 1 n , max

• > +
• ≤ + = χ .

• ≤ n p Since the number of leaves in a maximal subgraph is ,
• ≥ χ
• ≤ n
• n Consider the
• n -coloring c on ( )

  m C n n n m ..., , , ;

  max

  1

  max

  1

  p .

  for ≤

  1

  2

  Next, we determine the upper bound of ( )

  m C n n n m ..., , , ;

  p

  max

  1

  C n n n n m L m for .

  1

  2

  max

  by Corollary 1, ( ) ( ) , , 1 ..., , ;

  max n

  2

  as follows: a. Find the number of subgraphs

  1

  1 are colored by , ..., , 3 , 2 , 1 p respectively.

  1

  } ( ) { } . \

  ..., , 3 , 2 , 1 n

  max

  1 are colored by {

  K where p i ≤ ≤

  1 i n

  max ,

  c. Leaves in ,

  K x ∈ where p i ≤ ≤

  max , 1 n

  1 i n i

  max ,

  b. Vertices ,

  1 p i ≤ ≤ respectively.

  K where ,

  1 i n

  max ,

  of the subgraphs from left to right as ,

  K and denote it by p. Denote each

  max

  1

  1 for .

  1

  ( ) m

  , 1 n K be the maximum star subgraph of

  Theorem 2. Let max

  are called star subgraphs with the same distance.

  1

  K ,

  j n

  and

  K ,

  2

  i n

  then subgraphs

  K

  max , 1 n

  ,

  ≠ , are the centers of

  = with

x

and p p x x x

  , , , p j i x x d x x d

  Asmiati 1308 (2) ( ) ( )

  C n n n m ..., , , ;

  1 and p be the number of subgraphs . max

  

2

  1 ,

  

C n n n m ..., , , ;

  

( )

m

  First we determine the trivial lower bound of

  C n n n m L m Proof.

  1 n p if n n p if n

  2

  max max max max

  1 ..., , , ;

  2 , 1 ,

  ⎩ ⎨ ⎧

  , 1 n K Then for

  ( ) ( )

  is

  1

  2

  C n n n m ..., , , ;

  ( ) m

  max ≥ n the locating-chromatic number of non-homogeneous caterpillar

  2

  ,

  x c

• k
• k n
• n
• n

  2

  1

  2

  ..., , , ;

  m C n n n m

  We show that the color codes for all vertices in ( )

  j n

  2

  jl i il } . ..., ,

  , 1 ..., 2 , = 1 = | ≠ | l a c n l a c

  j i = x c x c then ( ) { } { ( ) ,

  versa, if ( ) ( ) ,

  j n then i x and j x should be given different colors. Vice

  } , ..., ,

  1

  = | = = | l a c n l a c jl i il

  1

  maximum star subgraph and ( ) { } { ( ) , , 1 ..., 2 ,

  

j i

= n n have the same distance from the

  with

  1

  K ,

  j n

  and

  1

  K ,

  i n

  i. If

  for ,

  max

  is colored by colors that correspond with ,

  

i

x and . j x

  p C and . q

  ≠ because they have different distances from

  ( ) v c Π

  A then ( ) u c Π

  p A and , q

  lie in different intervals, say

  1

  K ,

  j n

  and

  1

  K ,

  i n

  ≠ because their color codes differ in the ordinate of colors

  p are different. Let u , v , v u ≠

  Π Π

  then ( ) ( ) v c u c

  n n n j i

  max

  1

  and ( ) ( ) . v c u c =

  K V v ∈

  1 j n

  K V ( ) , ,

  1 i n

  ,

  u ( ) ,

  be the leaves, where ∈

  1 − k T respectively.

  1

  C

  1 max

  1

  from

  max n

  e. Let T = {all combinations ( )

  K

  1 p n

  ,

  A be an open interval after . max

  p k and 1 + p

  1 − ≤ ≤

  1

  K where ,

  1 ,

  color}, such that T =

  and ,

  1

  ,

  i n K max

  between

  i n A K be an open interval

  max

  ,

  1

  1 ,

  1 A be an open interval before

  d. Let

  On the Locating-chromatic Numbers of Non-homogeneous … 1309

  max

  { }

  K ,

  1 A or ,

  i n

  1 3 + ≤ ≤ p k then every vertex of

  k A where ,

  lies in interval ,

  1

  K ,

  i n

  h. If

  1 T respectively.

  is colored by colors that are associated with ,

  1

  K ,

  

2

A then every vertex of i n

  lies in interval

  1

  1

  K ,

  i n

  g. If

  1 in the interval as defined in item d.

  K ,

  i n

  f. Identify subgraph

  ∈ are color combination not containing the color i.

  T T T with T T i

  ..., , ,

  1 max

  2

• ≤ n
• If ,
• = =
• If

• If
• If one of
• If ( )

• ≤ n p are different, thus c is a locating-
• ≤ χ n n n n m C
• ≤ n p

  Figure 1. A minimum locating-coloring of C(9; 2, 3, 1, 1, 3, 1, 2, 3, 3).

• By Corollary 1, we have that ( ) ( ) .
• ≥ χ n n n n m C
• n colors are not enough. For a contradiction, assume that there exists a
• n -locating coloring c on
• > n
• > n

  max

  , 1 ..., , ;

  1

  max > n p .

  for

  1

  2

  m C n n n m ..., , , ;

  max

  Next, we show the lower bound of ( )

  1

  L m for .

  1

  2

  max

  coloring. So, ( ) ( ) , , 1 ..., , ;

  max

  1

  2

  1

  L m

  max

  max max n h a c n h a c

jh ih

  1

  such that ( ) { } { ( ) } . ..., , 2 , , 1 ..., 2 ,

  p there are two i, j , , j i ≠

  max

  1

  p Since ,

  1

  However, we will show that ( )

  1 for .

  2

  C n n n m ..., , , ;

  ( ) m

  max

  1

  max

  1

  for ,

  2

  1

  p A

  { } j i n n

  i x and . j x

  color codes differ in the ordinate of colors

  p C But, if they have the same distance, their

  ≠ because they have different distance to .

  Π Π

  but they do not have the same distance, then ( ) ( ) v c u c

  lie in the same interval, say

  max n say max n n i

  1

  K ,

  j n

  and

  1

  K ,

  i n

  Asmiati 1310

  , is ,

  = and ,

  C n n n m ..., , , ;

  K V x ,

  ( ) m

  From all the above cases, we see that the color codes for all vertices in

  

Π i Π

≠

  contains exactly one component of value 1. Thus, ( ) ( ) . v c x c

  Π

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

  Π

  1

∈ and v have the same color, then ( )

i x c

  j n i

  max n n j

  K

  1 i n

  ,

  not contained in .

  1

  K ,

  i n

  < then the color codes of u and v differ in color of leaves of

  = | = = |

  On the Locating-chromatic Numbers of Non-homogeneous … 1311 Therefore the color codes of and are the same, a contradiction. So,

  x x i j

  χ ≥ + > (

• C ( m ; n , n , ..., n ) ) n

  2 , for p n 1 .

  L

  1 2 m max max

  To show that χ ( C ( m ; n , n , ..., n ) ) ( ≤ n + 2 ) , consider the

  L

  1 2 m max

  locating-coloring c on C ( m ; n , n , ..., n ) as follows:

  1 2 m ( ) + • c l = n 2 .

  11 max

• The color of vertices x are:

  i

  1 if i odd , ⎧

  =

  c ( ) x

  ⎨

  i 2 if i even .

  ⎩

• Leaves, for n = 1 , c ( ) n =

  3 , whereas for n ≥ 2 , { ( ) c a | j = 1 , 2 ,

  i i i ij

  ⊆ + ..., n } , are colored by S { 1 , 2 , ..., n 1 } ( ) \ { c x } for any i.

  i max i

  Since there is only one vertex at the end of the longest path, which is

• colored by n

  2 , the color codes of all vertices are different. Therefore,

  max

c is the locating-chromatic coloring on C ( m ; n , n , ..., n ) , and so

  1 2 m

  χ ( ( + + C m ; n , n , ..., n ) ) ≤ n 2 , for p > n 1 .

  L

  1 2 m max max

Figure 2. A minimum locating-coloring of C(11; 3, 1, 1, 3, 1, 2, 3, 3, 1, 3, 1).

  

3. Locating-chromatic Number of Non-homogeneous

Firecracker Graphs

  A non-homogeneous firecracker graph, F is obtained by the

  n , ( k , k , ..., k )

  1 2 n

  concatenation of n stars S , i = 1 , 2 , ..., n by linking one leaf from each star.

  k i

  = | = = − Let

  V ( F ) { x , m , l i

  1 , 2 , ..., n ; j 1 , 2 , ..., k 2 } ,

  n , ( k , k , ..., k ) i i ij i

• n i x x

  1 max max max max

  number of leaves in the maximal subgraph is ,

  − ≤ k p Since the

  1 max

  L k k k n Proof. We determine the trivial lower bound of .

  F n

  1 k p if k k p if k

  2

  ..., , , ,

  1 , , 1 ,

  −

  = χ .

  − > − ≤ −

  ⎩ ⎨ ⎧

  

n is :

( )

  ≥

  

2

max

  F for

  2 1 n k k k n

  2 max

  k by Corollary 1, we have ( ( )

  1 and p be the number of subgraphs . max k

  1 max

  S where . 1 p i

  S and denote it by p . Denote each of the subgraphs from left to right by , max i k

  a. Find the number of subgraphs max

k

  1 as follows:

  2

  

F

..., , , ,

  k -coloring c on n k k k n

  −

  p Consider ( )

  ) ,

  − ≤ k

  1 max

  L k k k n for .

  F k n

  − ≥ χ

  1

  2

  1 ..., max , , ,

  S Then the locating - chromatic number of ( ), ..., , , ,

  2

  Asmiati 1312 and (

  {

  max k

  = then the subgraph

  2 1 max n k k k k

  i k j If { } , ..., , , max

  1 − =

  , 2 ..., 2 ,

  l m m x ij i i i ; ..., n } .

  , 2 , = 1 , |i

  i i ∪

  ..., , , ,

  − = |

  1

  1

  = { } , 1 ..., 2 ,

  1

  2

  F E ..., , , ,

  ( ) ) n k k k n

  

S is the maximum star subgraph of non-homogeneous firecracker

( ) .

  2 1 n k k k n

  F ..., , , ,

  (1) ( ) ( ) , , , m j m i

  S be the maximum star subgraph of ( ) n k k k n

  Theorem 3. Let max k

  S and j k S are called star subgraphs with the same distance.

  S then subgraphs i k

  ≠ x x x , are the centers of , max k

  = x x d x x d with x and p p

  S or (2) ( ) ( ) , , , p j i

  = x x d x x d with

m

x is the center of , max k

  ≠ = such that

  F If there are p subgraphs , max k

  , max n n n j i

  2 1 n j i k k k n k k F S S ⊂ where . 1 m j i ≤ ≠ ≤ If

  ..., , , ,

  Definition 2. Let ( ), ,

  1 p i ≤ ≤

  

S where .

  max i k

  left to right, is denoted by ,

  S then every subgraph, from

  ≤ ≤

  On the Locating-chromatic Numbers of Non-homogeneous … 1313 i

  b. Vertices x ∈ S , where 1 ≤ i ≤ p , are colored by 3 , 4 , 5 , ..., n , i k max

  2 , 3 , respectively. i

  ∈ ≤ ≤

  c. Vertices m S , where 1 i p , are colored by 1 , 2 , 3 , ..., p , i k max respectively. i

  d. Leaves in S , where 1 ≤ i ≤ p are colored by {

  1 , 2 , 3 , ...,

  k max

  ( k 1 )} ( ) \ { c x } .

  −

  max i i

  e. Let A be an open interval before S , A be an open interval

• 1 t

  1 k max

• t t

  1 between S and S , where 1 ≤ t ≤ p − 1 , and A be an

• p

  1 k k max max p open interval after S . k max

  f. Let T = {all combinations ( k −

  2 ) from ( k − 1 ) color}, such that

  max max T = { T , T , ..., T } with T ∈ T be combinations not containing

  1

  2

  1 k − i max color i .

  g. Identify subgraph S in the interval as defined in item . d k i

  h. If S lies in the interval A or A , then every vertex of S is k

  1 2 k i i colored by colors that correspond with T , respectively.

  1 3 ≤ k ≤ p

• i. If S lies in the interval A , where

  1 , then every vertex k k i of S , is colored by colors that correspond with T , respectively. k k

  1 − i

  =

  j. Let S and S , where k k have the same distance from the k k i j i j

maximum star subgraph. If c ( ) x = c ( ) x , then c ( ) m = c ( ) m .

i j i j

  Next, we show that the color codes for all vertices in F for n , k , k , ..., k

  1 2 n p ≤ k −

  1 , are different. Consider two distinct vertices u ∈

  V ( S ) and max k i v ∈ V ( S ) , where c ( ) u c ( ) v .

  =

  k j

  1314 Asmiati

  = ( ) • If = , then ( ) ≠ because their color codes

  k k k c u c v i j max Π Π differ in ordinate of colors m and m . i j

• If S and S lie in different intervals, say A and A , then c ( ) u

  k k p q Π i j

  ≠ c ( ) v because they have different distances from C and C .

  Π p q

• If S and S lie in the same interval, say A but they do not have

  k p k i j the same distance, then c ( ) u ≠ c ( ) v because they have different

  

Π Π

distance to C . But, if they have the same distance, their color codes p differ in ordinates m and m . i j

• If one of k , k } is k , say k = k and n k , then the

  <

  {

  i j max i max j max color codes of u and v differ in color of leaf of S , not contained in k i S . k j

• ∈

  If x V ( K ) and v lie in the same interval, say A , where c ( ) x i 1 , n k i j

  = c ( ) v , then c ( ) x and c ( ) v have different distance from C . So,

  Π i Π k c ( ) x c ( ) v . But, if they lie in different intervals, say A and A ,

  ≠

  i r s

  Π Π then c ( ) x c ( ) v because they have different distances from C

  ≠

  Π i Π r and C . s

  From all above cases, we see that the color codes for all vertices in F , for p ≤ k − 1 are different, thus c is a locating-coloring. n , k , k , ..., k max

  1 2 n So, χ ( F ) ≤ k − 1 , for p ≤ k − 1 .

  L n , ( k , k , ..., k ) max max

  1 2 n

  > k −

  We will show the lower bound for p 1 . By Corollary 1, we max

  χ ≥ −

  have that ( F ) k 1 . However, we will show that k L n , ( k , k , ..., k ) max max

  1 2 n

  −

  1 colors are not enough. For a contradiction that there exists a ( k − 1 ) - max locating coloring c on F for p > k − 1 . Since p > k n , ( k , k , ..., k ), max max

  1 2 n

  

[3] Asmiati and E. T. Baskoro, Characterizing all graphs containing cycle with the

locating-chromatic number 3, AIP Conf. Proc. 1450, 2012, pp. 351-357.

[4] Asmiati, H. Assiyatun and E. T. Baskoro, Locating-chromatic number of

amalgamation of stars, ITB J. Sci. 43A (2011), 1-8.

[5] E. T. Baskoro and Asmiati, Characterizing all trees with locating-chromatic

  max A k

  . otherwise , 2 \ , , 1 if

  = = − = |

  ⎩ ⎨ ⎧

  { }

  define: { ( ) } { }

  =

  1

  1 max max

  i

m c for i otherwise.

  2 =

  1 = k m c and ( )

  max

  i x c if i is even.

  =

  1 \ , 2 ..., 2 ,

  A k A i k k j l c i ij It is easy to verify that the color codes of all vertices are different.

  i x c if i is odd and ( )

  1 F k n

  Uttunggadewa, The locating-chromatic number of firecracker graphs, Far East J. Math. Sci. (FJMS) 63(1) (2012), 11-23.

  

[2] Asmiati, E. T. Baskoro, H. Assiyatun, D. Suprijanto, R. Simanjuntak and S.

  

References

[1] Asmiati, The locating-chromatic number of non-homogeneous amalgamation of

stars, Far East J. Math. Sci. (FJMS) 93(1) (2014), 89-96.

  max − > k p This completes the proof.

  1

  L k k k n ≤ χ for .

  2

  Therefore, c is a locating-chromatic coloring on ( )

  ..., max , , ,

  ( ) ) ,

  F and so (

  2 1 n k k k n

  ..., , , ,

  ,

  

3

  =

  On the Locating-chromatic Numbers of Non-homogeneous … 1315 , 1 − there are two i

  k l l c jl

  ≥ χ for .

  

L k k k n

  

1

F k n

  2

  ) ..., max , , ,

  Therefore, the color codes of i m and j m are the same, a contradiction. So, ( ( )

  − = |

  − > k

  1 max

  2 ,

  1 max k h l c ih { ( ) } . , 2 ...,

  

≠ such that ( ) { } = − = |

, 2 ..., 2 ,

  , , j i

  , j

  1 max

  p Next, we determine the upper bound of ( ) n k k k n

  1

  1 F k n

  1 as follows:

  2

  F ..., , , ,

  on ( ) n k k k n

  ≤ χ consider the locating-coloring c

  L k k k n

  2

  F ..., , , ,

  ..., max , , ,

  To show that ( ( )

) ,

  − > k p

  max

  1

  1 for .

  2

• ( )
• ( )
• If { } , ..., , 2 ,

  Asmiati 1316

[6] E. T. Baskoro and I. A. Purwasih, The locating-chromatic number for corona

product of graphs, AIP Conf. Proc. 1450, 2012, pp. 342-345.

  

[7] A. Behtoei and B. Omoomi, On the locating chromatic number of Kneser graphs,

Discrete Appl. Math. 159 (2011), 2214-2221.

[8] G. Chartrand, D. Erwin, M. A. Henning, P. J. Slater and P. Zhang, Graphs of

order n with locating-chromatic number n – 1, Discrete Math. 269(1-3) (2003),

  65-79.