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C om mun. Fac. Sci. U niv. A nk. Ser. A 1 M ath. Stat.

Volum e 68, N umb er 1, Pages 1229–1239 (2019) D O I: 10.31801/cfsuasm as.516089

ISSN 1303–5991 E-ISSN 2618-6470

http://com munications.science.ankara.edu.tr/index.php?series= A 1

ON b COLORING OF CENTRAL GRAPH OF SOME GRAPHS

M.KALPANA AND D.VIJAYALAKSHMI

Abstract. The b chromatic number of G, denoted by '(G), is the maximum kfor which G has a b coloring by k colors. A b coloring of G by k colors is a proper k-coloring of the vertices of G such that in each color class i there exists a vertex xihaving neighbors in all the other k 1color classes. Such a vertex xi is called a b dominating vertex, and the set of vertices fx1; x2: : : xkg is called a b dominating system. In this paper, we are going to investigate on the b chromatic number of Central graph of Triangular Snake graph, Sunlet graph, Helm Graph, Double Triangular Snake graph, Gear graph, and Closed Helm graph are denoted as C(Tn), C(Sn), C(Hn), C(DTn), C(Gn), C(CHn) respectively.

1. Introduction

Graph theory is the theory of graphs dealing with nodes and connections or vertices and edges. This subject has experienced explosive growth, due in large measure to its role as an essential structure underpinning modern applied mathe- matics. Con…gurations of nodes and connections has great diversity of applications.

They may represented physical networks, such as electrical circuits, roadways, or organic molecules. They are also used in representing less tangible interactions as might occur in ecosystem, sociological relationships, database, or in the ‡ow of control in a computer program. It is a fastest growing …eld in Mathematics mainly because of its applications in distinct areas. There are plenty of works have been done in di¤erent topics, like decomposition, domination, factoring, orienting, col- oring etc., in the past decades. A graph G = (V; E) is an ordered pair of two sets called V and E. Here elements of V are called vertices and elements of E are called edges. We consider the graphs here as undirected, …nite and neither have multiple edges nor loops. The order of G is denoted by n and the size is denoted by m. All graph terminologies are referred from J.A. Bondy and U.S.R. Murty [1].

Received by the editors: February 05, 2018; Accepted: June 26, 2018.

2010 Mathematics Subject Classi…cation. 05C15.

Key words and phrases. b coloring, b chromatic number, central graph.

Submitted via International Conference on Current Scenario in Pure and Applied Mathematics [ICCSPAM 2018].

c 2 0 1 9 A n ka ra U n ive rsity C o m m u n ic a tio n s Fa c u lty o f S c ie n c e s U n ive rs ity o f A n ka ra -S e rie s A 1 M a t h e m a t ic s a n d S ta t is t ic s

1229

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In Graph theory, Graph coloring is a well known area which is widely studied by many researchers. Various types of graph coloring and many open problems were discussed in the wonderful books [2; 7]. Graph coloring deals with the general and widely applicable concept of partitioning the underlying set of a structure into parts, each of which satis…es a given requirement. Here coloring means a vertex coloring of a graph. A k coloring of a graph G = (V; E) is a mapping C : V ! P where P is a set of k colors; thus k coloring is an assignment of k colors to the vertices of G. Usually, the set P of colors is taken to be f1; 2; : : : kg. A coloring C is proper if no two adjacent vertices are assigned the same color. Only loop less graphs admits proper coloring. The minimum k for which a graph G is k colorable is called its chromatic number, and denoted (G). If (G) = k, the graph G is said to be k chromatic.

The b chromatic number of G, denoted by '(G), is the maximum k for which G has a b coloring by k colors. A b coloring of G by k colors is a proper k- coloring of the vertices of G such that in each color class i there exists a vertex xi

having neighbors in all the other k 1 color classes. Such a vertex xi is called a b dominating vertex, and the set of vertices fx1; x2: : : xkg is called a b dominating system. The b coloring was introduced by R.W.Irving and D.F.Manlove in [6].

They proved that determining ' (G) is NP-hard in general and polynomial for trees.

A Triangular Snake [8] is obtained from a path x1; x2; : : : ; xn by joining xi and xi+1 to a new vertex yi for 1 i n. That is, every edge of a path is replaced by a triangle C3. The n-Sunlet graph Snis a graph [11] with cycle Cnand each vertex of the cycle attached to one pendent vertex. Each n-sunlet graph consists 2n nodes and 2n edges. A Helm Hn, n 3 is the graph [11] obtained from the Wheel Wn

by adding a pendent edge at each vertex on the rim of the Wheel Wn. The Central graph [9] of G, denoted by C (G) is obtained by subdividing each edge of G exactly once and joining all the non-adjacent vertices of G in C (G).

A Double Triangular Snake [12] is a graph formed by two Triangular Snakes having a common path. i.e., a Double Triangular Snake with k blocks is obtained from a path x1,x2,. . . xn by joining xi and xi+1 to two new vertices yi and zi for i = 1; 2; : : : n. A Gear graph[11] is obtained from the Wheel Wnby adding a vertex between every pair of adjacent vertices of rim of the Wheel Wn. A Closed Helm CHn is the graph obtained by taking a Helm Hn and adding edges between the pendent vertices. In this paper we have used algorithmic approach to prove the results of C(Tn), C(Hn) and C(DTn).

2. b-Coloring of C (Tn) 2.1. b-Coloring Algorithm of C (Tn).

Input: C(Tn), n 2.

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V fx1; x2; : : : xn; y1; y2; : : : yn; a1; a2; : : : an; b1; b2; : : : bn; c1; c2; : : : cng.

for i = 1 to n xi i;

end for for i = 1 to n yi i + n;

end for for i = 1 to n ai i + n;

end for for i = 1 to n bi i;

end for for i = 1 to n ci i + n + 1;

end for end procedure

Output: vertex colored C(Tn).

Theorem 1. For a Triangular Snake graph Tn, n 2, the b - chromatic number of Central Graph of Triangular Snake graph is 2n+1.

i:e:; ' [C (Tn)] = 2n + 1:

Proof. Let the vertex set of Triangular Snake graph as

V (Tn) = fxi: 1 i ng [ fyi: 1 i n + 1g

By the de…nition of Central graph, the edge joining xiand yi+1has been subdivided by the newly introduced vertex ai(1 i n) and the edge joining yi and yi+1 has been subdivided by the newly introduced vertex bi(1 i n).Denote the newly added vertex on the edge joining yi and xi as ci(1 i n). Then the vertex set of central graph of Triangular Snake graph is

V [C (Tn)] = fxi: 1 i ng [ fyi: 1 i n + 1g [ fai: 1 i ng [ fbi: 1 i ng [ fci : 1 i ng

The vertices of C (Tn) are colored as given in the algorithm 2:1.

By algorithm 3 (2n + 1) vertices fxi: 1 i ng [ fyi: 1 i n + 1g with color class

C = fi : 1 i 2n + 1g :

By algorithm the set of vertices fxi: 1 i ng [ fyi: 1 i n + 1g with cardi- nality 2n+1 has the color classes

C [fxi : 1 i ng [ fyi: 1 i n + 1g] = fi : 1 i 2n + 1g

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Since

jfxi: 1 i ng [ fyi: 1 i n + 1gj = 2n + 1 = jfi : 1 i 2n + 1gj ; the vertices of fxi: 1 i ng[fyi: 1 i n + 1g receive distinct colors. C [N(xi)]

and C [N (yi)] have 2n distinct colors for each i. It implies that the coloring is b coloring. To prove it is maximum, let us suppose that '[C(Tn)] > 2n + 1. Then there exists at least 2n + 2 vertices of degree 2n + 1. The C[Tn] has 2n + 1 ver- tices of degree 2n, and other vertices are of degree 2. i.e., 3 no vertex of degree 2n + 2, which is a contradiction to the fact that '[C(Tn)] > 2n + 1. Therefore ' [C (Tn)] 2n + 1: Hence ' [C (Tn)] = 2n + 1:

3. b Coloring of C (Sn)

Theorem 2. For a Sunlet graph Sn, n 3, the b - chromatic number of Central Graph of Sunlet graph is (3n 1)=2 for odd n and n + (n=2) for even n.

i:e:; '[C(Sn)] = 8>

<

>:

(3n 1)=2; for n is odd n + (n=2); for n is even:

Proof. Let

V (Sn) = fxi: 1 i ng [ fyi: 1 i ng :

By the de…nition of Central graph, the edge joining xi and xi+1(1 i n) has been subdivided by the newly introduced vertex ai(1 i n 1), the remaining edge xnx1has been subdivided by the vertex an and the edge joining xiand yihas been subdivided by the newly introduced vertex bi(1 i n). Then the vertex set of central graph of Sunlet graph is

V [C (Sn)] = fxi: 1 i ng [ fyi: 1 i ng [ fai: 1 i ng [ fbi: 1 i ng

Now let us assign the b-coloring to the vertex set of [C (Sn)] by using the following function,

f : V [C (Sn)] ! Ci: 1 i n:

Case (i): When n is odd

f (xi) = f (xi+1) = Cn+i; 1 i n 3;

f (xn 2) = f (xn 1) = f (xn) = C(3n 1)=2; n 2 i n:

Next consider the vertex set of yi(1 i n). The coloring functions of yi are, f (yi) = Ci; 1 i n:

The coloring functions of newly introduced vertex set ai(1 i n) are, f (ai) = Ci; 1 i n:

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and bi(1 i n) are,

f (bi) = C1+i; 1 i n:

Then the vertices of C(Sn) are colored by the above coloring process. By the above coloring 3 (3n 1)=2 vertices of fxi: 1 i ng [ fyi : 1 i ng with cardinality 2n has the color class

C = fi : 1 i (3n 1)=2g : N [xi] = fx1; x2; : : : ; xn 2; xn 1; xng C [N [xi]] = fn + i; : : : ; (3n 1)=2g

N [yi] = fyi: 1 i ng : C [N [yi]] = fi : 1 i ng Then

C [N [xi]] [ C [N[yi]] = (3n 1)=2:

It implies the b coloring. To prove it is maximum. Let us assume that, '[C(Sn)] >

(3n 1)=2. The (3n)=2 color does not have neighbors of other colors. Then it does not satisfying b coloring, which is a contradiction to the fact that '[C(Sn)] >

(3n)=2. Therefore, '[C(Sn)] (3n + 1)=2. Hence '[C(Sn)] = (3n + 1)=2.

Case (ii): When n is even

f (xi) = f (xi+1) = Cn+i; 1 i n 2;

f (xn 1) = f (xn) = Cn+(n=2); n 1 i n:

Next consider the vertex set of yi(1 i n). The coloring functions of yi are, f (yi) = Ci; 1 i n:

The coloring functions of newly introduced vertex set ai(1 i n) are, f (ai) = Ci; 1 i n:

and bi(1 i n) are,

f (bi) = C1+i; 1 i n:

Then the vertices of C(Sn) are colored by the above coloring process. By the above coloring 3 n + (n=2) vertices of fxi: 1 i ng [ fyi: 1 i ng with cardinality 2n has the color class

C = fi : 1 i n + (n=2)g N [xi] = fx1; x2; : : : ; xn 1; xng : C (N [xi]) = fn + i; : : : ; n + (n=2)g

N [yi] = fyi: 1 i ng : C (N [yi]) = fi : 1 i ng

Then C (N [xi]) [ C (N[yi]) = n + (n=2): It implies the b-coloring. To prove it is maximum. Let us assume that, '[C(Sn)] > n + (n=2). The n + (n=2) + 1 color is not adjacent to all other colors. Then it does not satisfy b coloring, which is

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a contradiction to the fact that '[C(Sn)] > n + (n=2). Therefore, '[C(Sn)]

n + (n=2). Hence '[C(Sn)] = n + (n=2).

4. b Coloring of C (Hn) 4.1. b-Coloring Algorithm of C (Hn).

Input: C (Hn) ; n 3:

V fx; x1; x2; : : : xn; y1; y2; : : : yn; a1; a2; : : : an; b1; b2; : : : bn; c1; c2; : : : cn:g x n + 1;

for i = 1 to n if i = 1; 2; n xi n + 2;

else xi i;

end for for i = 1 to n yi i;

end for for i = 1 to n if i = 1, ai i;

else

ai n + 1;

end for for i = 1 to n bi n + 1;

end for for i = 1 to n if i = 1; 2; n ci n;

else

ci n + 2;

end for end procedure

Output: vertex colored C (Hn) :

Theorem 3. For a Helm graph Hn, n 3; the b-chromatic number of Central Graph of Helm graph is n+2.

i:e:; ' [C (Hn)] = n + 2:

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Proof. Let us consider the vertex set of helm graph

V (Hn) = x [ fxi: 1 i ng [ fyi: 1 i ng :

By the de…nition of Central graph, the edge joining xiand xi+1has been subdivided by the newly introduced vertex ai(1 m n 1), the remaining edge xnx1 has been subdivided by the vertex anand the edge joining xiand yihas been subdivided by the new vertex bi(1 i n). Denote the newly added vertex on the edge joining xi and x as ci(1 i n) : Then the vertex set of central graph of Helm graph is

V [C (Hn)] = x [ fxi: 1 i ng [ fyi: 1 i ng [ fai: 1 i ng [ fbi: 1 i ng [ fci: 1 i ng :

The vertices of C(Hn) are colored as given in the algorithm 4:1. By algorithm 3 n+2 vertices fxg [ fx1g [ fyi: 1 i ng with color class C = (i : 1 i n + 2). By algorithm the set of vertices fxg [ fx1g [ fyi: 1 i ng with cardinality n+2 has the color class

C[ x [ x1[ yi: 1 i n] = (i : 1 i n + 2)

and the vertices of x [ x1[ yi : 1 i n receive distinct colors. This implies the fact that, the coloring is b-coloring. To prove it is maximum. Let us suppose that '[C(Hn)] > n + 2: Then there must be at least n + 3 vertices having degree n + 2 in C (Hn), with distinct colors and also adjacent to all other colors. Then only we can assign n + 3 colors to the vertex set of ' [C (Hn)]. Here the vertex set fyi : 1 i ng and x in C (Hn) forms a clique of order n. If we assign the n + 3 color, then it does not satisfy the b coloring condition. Therefore we can assign n+1 colors to the clique and n + 2 color to the vertex x1. Which produce b coloring. Therefore ' [C (Hn)] n + 2: Hence ' [C (Hn)] = n + 2:

5. b Coloring of C (DTn) 5.1. b Coloring Algorithm of C (DTn).

Input: C (DTn) ; n 1:

V fx1; x2; : : : xn; y1; y2; : : : yn; z1; z2; : : : zn; a1; a2; : : : an; b1; b2; : : : bng.

V fc1; c2; : : : cn; d1; d2; : : : dn; e1; e2; : : : en:g for i = 1

x1 2(n + 1);

end for for i = 2 x2 2n + 1;

end for for i = 3 to n

xi 2k; k = 1; 2; : : : ; n ; end for

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for i = 1 to n yi i;

end for for i = 1 to n zi i + n;

end for for i = 1 to n

ai k; k = 1; 3; : : : n;

end for for i = 1 to 2 bi i + 2n;

end for for i = 3 to n bi 2n + 1;

end for for i = 1 to 2 if i = 1 ci 2(n + 1);

else ci 2n + 1;

end for for i = 3 to n ci 2(n + 1);

end for for i = 1 to 2 if i = 1 di 2n + 1;

else

di 2(n + 1);

end for for i = 3 to n di 2n + 1;

end for for i = 1 to 2 if i = 1 ei 2(n + 1);

else

ei 2n + 1;

end for for i = 3 to n ei 2(n + 1);

end for

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end procedure

Output: vertex colored C (DTn) :

Theorem 4. For a Double Triangular Snake graph DTn, n 2, the b - chromatic number of Central Graph of Double Triangular Snake graph is 2(n+1).

i:e:; ' [C (DTn)] = 2(n + 1):

Proof. Let DTn be a Double Triangular Snake with i blocks on n vertices. Let V1 = fxi: 1 i n + 1g

V2 = fyi: 1 i ng V3 = fzi: 1 i ng The vertex set of Double Triangular Snake is

V (DTn) = V1[ V2[ V3: E1 = fxixi+1: 1 i ng E2 = fyixi: 1 i ng E3 = fyixi+1: 1 i ng E4 = fzixi: 1 i ng E5 = fzixi+1: 1 i ng The edge set of Double Triangular Snake is

E(DTn) = E1[ E2[ E3[ E4[ E5:

By using central graph de…nition, let us subdivide each edge of DTnby introducing a new vertex between each edge. The edge xixi+1(1 i n), has been subdivided by the vertex ai(1 i n). The edges yixi and yixi+1 are subdivided by the vertex bi(1 i n) and ci(1 i n) respectively. Denote the newly introduced vertex of zixi and zixi+1 as di(1 i n) and ei(1 i n). Then the vertex set of C(DTn) is

V [C(DTn)] = fxi: 1 i n + 1g [ fyi: 1 i ng [ fzi : 1 i ng [ fai : 1 i ng [ fbi: 1 i ng [ fci: 1 i ng [ fdi: 1 i ng [ fei: 1 i ng

The vertices of C(DTn) are colored as given in the algorithm 5:1.

By algorithm 3 2(n + 1) vertices

fxi: 1 i n + 1g [ fyi : 1 i ng [ fzi: 1 i ng [ fai: 1 i ng [ fbi: 1 i ng [ fci: 1 i ng [ fdi: 1 i ng [ fei: 1 i ng

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with color class C = f1 i 2(n + 1)g. The set of vertices fxi: 1 i 2g [ fyi: 1 i ng [ fzi: 1 i ng have cardinality 2(n + 1) with color class

C[fxi: 1 i 2g [ fyi: 1 i ng [ fzi: 1 i ng] = fi : 1 i 2(n + 1)g Since

jfxi: 1 i 2g [ fyi: 1 i ng [ fzi: 1 i ngj

= 2(n + 1) = jfi : 1 i 2(n + 1)gj the vertices of fxi: 1 i 2g [ fyi: 1 i ng [ fzi: 1 i ng receive distinct colors. C[N (xi)], C[N (yi)] and C[N (zi)] have 2(n + 1) distinct colors for each i. It implies that, the coloring is b coloring. To prove it is maximum. Let us suppose that '[C(DTn)] > 2(n + 1).

For assigning 2(n+2) colors to C(DTn) we need 2(n+3) vertices of degree 2(n+2), all are having distinct colors and adjacent to all other colors. Here in C(DTn) we have a clique formed by the vertex sets fxi: 1 i ng [ fyi: 1 i ng. so that we can assign 2n colors to the vertex sets of yi and zi. In xi the vertex set fxi: 1 i n + 1g having degree 3n, so we can assign two more colors. Therefore '[C(DTn)] 2(n + 1). Hence '[C(DTn)] = 2(n + 1).

6. b Coloring of C (Gn)

Theorem 5. For a Gear graph Gn, n 3, the b - chromatic number of Central Graph of Gear graph is n + b(2n 1)=2c for n is 3 and 2(n 1) for n 4.

i:e:; '[C(Gn)] = 8>

<

>:

n + b(2n 1)=2c ; for n = 3

2(n 1); for n 4:

7. b Coloring of C (CHn)

Theorem 6. For a Closed Helm graph CHn, n 3, the b - chromatic number of Central Graph of Closed Helm graph is n + dn=2e :

i:e:; '[C(CHn)] =

(n + dn=2e ; for n is odd:

References

[1] Bondy, J.A. and Murty, U.S.R., Graph Theory with Applications. MacMillan, London, 1976:

[2] Chartrand, G. and Zhang, P., Chromatic Graph Theory, Chapman and Hall, CRC, 2009:

[3] E¤antin, B., The b-chromatic number of power graphs of complete caterpillars, Journal of Discrete Mathematical Sciences and Cryptography, 8 (2005) 483-502:

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[4] E¤antin, B. and Kheddouci, H., The b-chromatic number of some power graphs, Discrete Mathematics and Theoretical Computer Science, 6 (2003) 45-54.

[5] Gallian, A., A dynamic survey of Graph Labeling, 2009:

[6] Irving, R.W. and Manlove, D.F., The b-chromatic number of a graph, Discrete Applied Mathematics, (1999) 127-141, 91.

[7] Jensen, T.R. and Toft, B., Graph Coloring Problems. Wiley-Interscience, 2009.

[8] Somasundaram,S. Sandhya, S.S. and Viji, S.P., On Geometric Mean Graphs, International Mathematical Forum, 10(3), (2009) 115-125.

[9] Vernold, Vivin J., Harmonious Coloring of Total graphs, n- leaf, Central graphs and circum- detic graphs, Ph.D Thesis, Bharathiyar University, Coimbatore, India, 2007.

[10] Vernold, Vivin J., Venkatachalam, M. and Akbar, Ali M.M., A Note on Achromatic Coloring of Star Graph Families, Filomat, 23 (3) (2009) 251-255.

[11] Vernold, Vivin J. and Venkatachalam, M., On b- chromatic number of Sunlet and Wheel Graph Families, Journal of the Egyptian Mathematical Society, 23(2) (2015) 215-218:

[12] Yue, Xi, Yuansheng, Yang and Liping, Wang, On Harmonious Labeling of the Double Trian- gular Snake, Indian Journal of Pure and Applied Mathematics, (2008) 177-184, 39(2).

Current address : M.Kalpana : Department of Mathematics, Kongunadu Arts and Science College Coimbatore - 641 029, Tamilnadu India

E-mail address : kalpulaxmi@gmail.com

ORCID Address: http://orcid.org/0000-0002-3303-6590

Current address : D.Vijayalakshmi : Department of Mathematics, Kongunadu Arts and Science College Coimbatore - 641 029, Tamilnadu India

E-mail address : vijikasc@gmail.com

ORCID Address: http://orcid.org/0000-0002-8925-1134

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