*Permanent address: Mathematics Departement, Faculty of Mathematics and Natural Sciences, Universitas Lampung, Jl. Brojonegoro no.1 Bandar Lampung, Lampung.
Received January 11th, 2010, Revised July 14th,2010, Accepted for publication July 28th, 2010.
Locating-Chromatic Number of Amalgamation of Stars
Asmiati*, H. Assiyatun & E.T. Baskoro Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jl. Ganesa 10 Bandung.
Email: [email protected]
Abstract. Let G be a connected graph and c a proper coloring of G . For 1,2, ,
i k define the color class C as the set of vertices receiving color i i. The color code c( )v of a vertex v in G is the ordered k -tuple ( ( ,d v C1),, ( ,d v Ck)) where d v C( , i) is the distance of v to C . If all distinct i vertices of G have distinct color codes, then c is called a locating-coloring of G . The locating-chromatic number of graph G , denoted by L( )G is the smallest k such that G has a locating coloring with k colors. In this paper we discuss the locating-chromatic number of amalgamation of stars Sk m, . Sk m, is obtained from k copies of star K1,m by identifying a leaf from each star. We also determine a sufficient condition for a connected subgraph HSk m, satisfying L( )H L(Sk m, ).
Keywords: amalgamation of stars; color code; locating-chromatic number.
1 Introduction
Let G be a finite, simple, and connected graph. Let c be a proper coloring of a connected graph G using the colors 1, 2, for some positive integer k , , k where ( )c u c v( ) for adjacent vertices u and v in G . Thus, the coloring c can be considered as a partition of V G into color classes (independent sets) ( )
1, 2,..., k
C C C , where the vertices of C are colored by i for 1 ii k. The color code c( )v of a vertex v in G is the ordered k -tuple ( ( ,d v C1),, ( ,d v Ck)) where d v C( , i)min{ ( , ) |d v x xCi}for 1 i k. If all distinct vertices of G have distinct color codes, then c is called a locating-coloring of G . A minimum locating-coloring uses a minimum number of colors and this number is called the locating-chromatic number of graph G , denoted by L( )G .
The following definition of an amalgamation of graphs is taken from [3]. For 1, 2, ,
i , let k G be a graph with a fixed vertex i v . The amalgamation oi Amal(G vi, oi) is a graph formed by taking all the G ’s and identifying their i fixed vertices. In this paper, we consider the amalgamation of stars. More precisely, for i1, 2,...,k, let 1, , 1
i ni i
G K n where v be any leaf (a vertex of oi degree 1) of 1,
ni
K . We denote the amalgamation of k stars 1,
ni
K by
1 2
,( , ,...,k)
k n n n
S ,k2. We call the identified vertex as the center (denoted by x ), the vertices of distance 1 from the center as the intermediate vertices (denoted by li ;i1, 2,...,k), and the j -th leaf of the intermediate vertex l by i
( 1, 2,..., 1)
lij j m . In particular, when nim, m1 for all i , we denote the amalgamation of k isomorphic stars K1,m by Sk m, .
The locating-chromatic number was firstly studied by Chartrand et al. [1]. They determined the locating-chromatic numbers of some well-known classes such as paths, cycles, complete multipartit graphs and double stars. Apart of paths and double stars, the locating-chromatic number of other trees are also considered by Chartrand et al. [2]. They constructed a tree of order n5 with the locating- chromatic number k , where k{3, 4,...,n2, }n . They also showed that no tree on n vertices with locating-chromatic number n1.
Based on the previous results, locating-chromatic number of amalgamation of stars have not been studied. Motivated by this, in this paper we determine the locating-chromatic number of amalgamation of stars.
Beside that, we also discuss the monotonicity property of the locating- chromatic number for the class of amalgamation of stars. Clearly, the locating- chromatic number of a star K1,n is n1, for any n (since all vertices must have different color codes). Since any connected subgraph H of K1,n is also a star with small size, then we clearly have L( )H L(K1,n). However in general for any connected subgraph HG, the locating-chromatic number of H may not be necessarily smaller or equal to the locating-chromatic number of G . In this paper, we also investigate the monotonicity property of the locating- chromatic number for amalgamation of stars, Sk m, . We derive a sufficient condition for a connected subgraph H Sk m, satisfying L(H)L(Sk m, ).
The following results were proved by Chartrand et al. in [1] . The set of neighbours of a vertex v in G is denoted by N v . ( )
Theorem 1.1. Let c be a locating-coloring in a connected graph G . 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 Gis a connected graph containing a vertex adjacent to k leaves of G , then L( )G k 1.
2 Main Results
We first prove some lemmas regarding the properties of locating-chromatic number of amalgamation of stars. From now on Sk m, denotes the amalgamation of k isomorphic stars K1,m.
Lemma 2.1. For k2,m2, let c be a proper coloring of Sk m, , using at least m colors. The coloring c is a locating-coloring if and only if
( )i ( ),n
c l c l in implies { ( ) |c lij j1, 2,...,m1} and { ( ) |c lnj j1, 2,...,m1}
are distinct.
Proof. Let P{ ( ) |c lij j1, 2,...,m1} and Q{ ( ) |c lnj j1, 2,...,m1}. Let c be a locating-coloring of Sk m, , k2,m2 using at least m colors and
( )i ( )n
c l c l , for some in. Suppose that PQ. Because ( , )d l ui d l u( , )n for every u V \ { |
lij j1, 2,...,m 1} {lnj| j1, 2,...,m1}
then the color codes of l and i l will be the same. So c is not a locating-coloring, a contradiction. n Therefore PQ.Let be a partition of V G into color classes with |( ) | m. Consider ( )i ( ),n
c l c l in. Since PQ, there are color x and color y such that (xP x, Q) and (yP y, Q). We will show that color codes for every
( k m, )
v V S is unique.
Clearly, c l( )i c l( )n because their color codes differ in the x th- ordinate and y th-ordinate.
If ( )c lij c l( )ns , for some liln, we will show that c l( )ij c l(ns). We divide into two cases.
Case 1. If ( )c li c l( )n then by the premise of this theorem, PQ. So ( )ij (ns)
c l c l .
Case 2. Let c l( )i r1 and c l( )n r2, with r1r2. Then c l( )ij c l(ns) because their color codes are different at least in the r th-ordinate and 1
r th-ordinate. 2
If ( )c li c l( )nj , li ln, then c l( )i contains at least two components of value 1, whereas c l( )nj contains exactly one component of value 1.
Thus c l( )i c l( )nj .
If c x( )c l( )ij , then color code of c( )x contains at least two components of value 1, whereas c l( )ij contains exactly one component of value 1. Thus c( )x c l( )ij .
From all above cases, we see that the color code for each vertex in Sk m, is unique, thus c is a locating-coloring.
Lemma 2.2. Let c be a locating coloring of Sk m, using m a colors and
( ) ( 1) 1 , 0
1
H a m a m a a
m
. Then kH a( ).
Proof. Let c be a locating-coloring of Sk m, using m a colors. For fixed i , let ( )c l be a color of intermediate vertex i l ,then the number color combina-i tions can be used by { |lij j1, 2,...,m1} is 1
1 m a
m
. Because one color is used for coloring the center x , there are (m a 1) colors for l , for every i
1, 2,...,
i k. By Lemma 2.1, the maximum number of k is
( 1) 1 ( )
1 m a
m a H a
m
. So kH a( ).
The main result of this paper concerns about locating-chromatic number of
,
Sk m.
Theorem 2.1. For a0,k2,m2, let ( ) ( 1) 1 1 m a H a m a
m
. Then,
,
2 (0), 3,
( )
( 1) ( ), 1.
L k m
m for k H m
S m a for H a k H a a
Proof. First, we determine the trivial lower bound. By Corollary 1.1, each vertex l is adjacent to (i m1)leaves, for i1, 2,...,k. Thus, L(Sk m, )m. Next, we determine the upper bound of L(Sk m, ) for 2 k H(0) m 1. Let
c be a coloring of V S( k m. ) using m colors. Without loss of generality, we can assign ( ) 1c x and ( )c li i 1 for i1, 2,...,k. To make sure that the leaves will have distinct color code, we assign { |lij j1, 2,...,m1} by {1, 2,..., }\{m i1} for any i . Then, by Lemma 2.1, we have that c is a locating-coloring. Thus L(Sk m, )m.
Next, we shall improve the lower bound for the case of k such that
( 1) ( ), 1
H a k H a a . Since kH a( 1) then by Lemma 2.2, ( , )
L Sk m m a
. On the other hand if kH a( ) then by Lemma 2.2,
( , ) 1
L Sk m m a
. Thus L(Sk m, ) m a if H a( 1) k H a( ).
Next, we determine the upper bound of L(Sk m, ) for H a( 1) k H a( ), 1
a . Without loss of generality, let c x( ) 1 and color the intermediate vertices l by 2,3,..., m ai in such a way that the number of the intermediate vertices receiving the same color t does not exceed 1
1 m a
m
, for any t . We are able to do so because H a( 1) k H a( ). Therefore, if ( )c li c l( )n , in then we can manage { ( ) |c lij j1, 2,...,m 1} { ( ) |c lnj j1, 2,...,m1}. By Lemma 2.1, c is a locating-coloring on Sk m, . So L(Sk m, ) m a for
( 1) ( )
H a k H a .
The following figures show minimum locating-colorings on S4,6 and S9,3 .
1 1
3 4 5
6
2
6 5 2
3 4 5
1
1
3 2 5 6 2 1
3
4 6
Figure 1 A minimum locating-coloring of S4,6.
4
3
4 1
1 2
1 3
2
2
3 4
4
2 1 4
4
1 2
2
3 3
1 3
1
2 4 3
Figure 2 A minimum locating-coloring of S9,3.
Next, we discuss the monotonicity property of locating-chromatic number for the amalgamation of stars.
Theorem 2.2 If 2 k m 1, then L( )G L(Sk m, ) for every GSk m, and GK1,m.
Proof. Let c be a minimum locating-coloring of Sk m, obtained from Theorem 2.1. Let G be any connected subgraph of Sk m, . Define a coloring con G by
preserving colors used in Sk m, for the corresponding vertices, namely ( ) ( )
c v c v if v is the corresponding vertex of v in Sk m, . We show that c is a locating-coloring of G .
If there exist l , i l such that { ( ) |n c l ij j1, 2,..., } { ( ) |r c l nj j1, 2,..., }r , with 1 r m 1, then color codes of lij and lnj for every j1, 2,3,...,m is unique because c l( )i c l( )n for every liln. If c l( )i c l( )nj c x( ), then the first component of c l( )i has value 1, whereas for c l( )nj it has value 2. So color code of l and i lnj are different. Next, if ( )c x c l( )nj , GP3 then their color codes are different because ( )c l i c l( )n for every liln. For the case GP3,
( )3
viV P for each i is colored by 1, 2, and 3 respectively. Because the color codes for every v V G ( ) is unique, then c is a locating-coloring of G . So
( ) ( , )
L G L Sk m
for every GSk m, , GK1,m.
Let Sk n n,(1,2,...,nk) Sk m, . Define A{ |i ni 1}. For km, we must restrict subgraphs of Sk m, so that satisfy monotonicity property.
Theorem 2.3 If km and AL(Sk m, ) 1 then
1 2
,( , ,..., ) ,
( ) ( )
L Sk n n nk L Sk m
.
Proof. Let km and from Theorem 2.1, we have that L(Sk m, ) m a for
( 1) ( ), 1
H a K H a a . Let
1 2
,( , ,...,k)
k n n n
GS be any subgraph obtained from
,
Sk m with 1 ni m. If 2 ni m for each i , then color vertices of G follow the proof of Theorem 2.1. Clearly, the coloring of G is a locating-coloring.
Otherwise, we have ni1 for some i , and so |A| 1 . If AL(Sk m, ) 1 , then the center x is given color 1, liA for each i is colored by 2,3,...,L(Sk m, ), respectively and the colors for the other vertices follow the proof of Theorem 2.1. Observe that the color codes of l for each i liA has value 1 in the 1th- ordinate, 0 in the i th-ordinate, and 2 otherwise, these color codes are unique.
For the remaining of the vertices, the color codes are also unique as proven in Theorem 2.1. As the result, the coloring of G is a locating-coloring. So
1 2
,( , ,..., ) ,
( ) ( )
L Sk n n nk L Sk m
.
References
[1] Chartrand, G., Erwin, D., Henning, M.A., Slater, P.J. & Zang. P., The locating-chromatic number of a graph, Bull. Inst. Combin. Appl., 36, pp.
89-101, 2002.
[2] Chartrand, G., Erwin, D., Henning, M.A., Slater & P.J., Zang. P., Graph of order n with locating-chromatic number n1, Discrete Mathematics, 269, pp. 65-79, 2003.
[3] Carlson, K., Generalized books and C -snakes are prime graphs, Ars m Combin., 80, pp. 215-221, 2006.
