Star Coloring of Cartesian Product of Complete Bipartite Graphs, Double Star Graphs with Complete Bipartite Graphs

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Star Coloring of Cartesian Product of Complete Bipartite Graphs, Double Star Graphs with Complete Bipartite Graphs

P.Hemalatha

¹

and S.N.Subhathra*

1Assistant Professor, Department of Mathematics, Vellalar College for Women, Erode, Tamilnadu, India

2Research Scholar, Department of Mathematics, Vellalar College for Women, Erode, Tamilnadu, India

Abstract

Let Km,n and D(K1,r ,K1,s) denotes a complete bipartite graph with m and n vertices and a double star graph on r and s vertices respectively. Let G1□G2 denotes the cartesian product of two graphs G1 and G2. In this paper, the star coloring of graphs such as Km,n□Km,n and D(K1,r,K1,s)□Km,n have been studied and the star chromatic number χs is obtained for such graphs. It is proved that for given positive integers m, n, r, s ≥ 2 ,

(i) χs(Km,n□Km,n) = 2 min {m, n} + 1 . (ii) χs(D(K1,r, K1,s)□Km,n) = min{m, n} + 3 .

Keywords: Cartesian Product of Graphs, Complete Bipartite Graphs, Double Star Graphs, Star Coloring, Star Chromatic Number.

1 Introduction

All graphs considered here are finite simple undirected connected graphs.

A star coloring [6] of a graph G is a proper coloring of G with the condition that no path on 4 vertices is 2-colored. A k-star coloring of a graph G is a star coloring of G using atmost k colors. If G[X,Y] is simple and every vertex in X is joined to every vertex in Y, then G is called a complete bipartite graph [1] denoted by Km,n , if lXl=m and lYl=n. A double star graph D(K1,r, K1,s) [8] is obtained by joining the centre u of the star K1,n and the center v of another star K1,m to a vertex w by an edge, with number of vertices n+m+2 and the number of edges n+m+1. The cartesian product [4] of two graph G and H , denoted by G□H , is a graph with vertex set V( G□H ) = V (G) × V (H)

= {(g, h)|g ∈ G, h ∈ H}. The edge set of G□H consists of all pairs [(g1, h1), (g2, h2)] of vertices with [g1, g2] ∈ E(G) and h1 = h2 or g1 = g2 and [h1, h2] ∈ E(H) .

The star coloring of graphs was first introduced by Gru¨ nbaum [3] in 1973 and he[3] discussed the proper coloring in which P4 is not bicolored and this coloring is known as star coloring. Later, Yu- ping Tsao et al[10] has given acyclic chromatic index of cartesian product of graphs. In recent decades, G.Fertin et al[2] have given the star coloring of some families of graphs such as trees, cycles, complete bipartite and obtained the bounds for the star chromatic number of other families of graphs, such as planar graphs, hypercubes, etc. N.Ramya[7] has compared the bounds of proper coloring, acyclic coloring and star coloring of Wheel Graphs. Recently, the star coloring of the cartesian product of paths and cycles have been investigated by Tianyong Han et al[9] and they have used k- star coloring for SAT(satisfiability test). The star coloring of prism graph, lollipop graph, barbell graph, Windmill graph, etc., have been discussed by [6]. Hemalatha and Subhathra[5] has given the star chromatic number of cartesian product of paths and cycles with complete bipartite graphs. In this paper, the star chromatic number of the cartesian product of complete bipartite graphs and double star graphs with complete bipartite graphs have been obtained.

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2 Star Coloring of Cartesian Product Of Graphs

2.1

Star Coloring of Km,n□Km,n

In this section, the star coloring of the cartesian product of complete bipartite graph Km,n with the complete bipartite graph Km,n for given positive integers m, n ≥ 2 is discussed.

Theorem 2.1. For given positive integers m, n ≥ 2, χs(Km,n□Km,n) = 2 min{m, n} + 1 .

Proof. Let G = Km,n□Km,n . From the definition of G it is clear that G has m + n layers of m+n vertices. Let aij denotes a vertex in the i^th layer of the j^th column.

i.e.,V( Km,n□Km,n ) = [aij ](m+n)×(m+n) .

Let m ≥ n . If c(aij) denotes the coloring of the vertex aij , then the star coloring of the vertices aij , i, j = 1, 2, 3, ..., m, m + 1, m + 2, ..., m + n can be dealt in 4 cases:

Case(i): i = 1, 2, 3, ..., m and j = 1, 2, 3, ..., m

Case(ii): i = m + 1, m + 2, m + 3, ..., m + n and j = 1, 2, 3, ..., m Case(iii): i = 1, 2, 3, ..., m and j = m + 1, m + 2, m + 3, ..., m + n

Case(iv): i = m + 1, m + 2, m + 3, ..., m + n − 1, m + n and j = m + 1, m + 2, m + 3, ..., m + n For m > n , we have the star coloring of V (Km,n□Km,n) as given in Table 2.1.1:

Vertices aij

i J c(aij)

Case(i) 1, 2, 3, ..., m 1, 2, 3, ..., m 1

Case(ii) m + 1, m + 2, m + 3, ..., m + n

1, 2, 3, ..., m c(aij) =

 



 

























m j

n m i if n

m j

m i if

m j

m i if

, ...

, 3 , 2 , 1 , ,

1 . . .

, ...

, 3 , 2 , 1 , 2 ,

3 , ...

, 3 , 2 , 1 , 1 ,

2

Vertices aij

i j c(aij)

Case(iii)

1, 2, 3, ..., m m + 1, m + 2, m + 3,

..., m + n c(aij) =



























n m j m i

if n

m j m i

if n

m j m i

if n

, , ...

, 3 , 2 , 1 , 1 2 . . .

2 ,

, ...

, 3 , 2 , 1 , 3

1 ,

, ...

, 3 , 2 , 1 , 2

Case(iv)

m +1, m +2, m +3, ..., m +n −1, m +n

















 

n m i for m j

n m m

m i for m j a i

c _ij

,

1 ,

...

, 2 , 1 ) ,

(

except for i + j = km + n + 1 for some positive integer k ≥ 2

If i + j = km + n + 1 , where k ≥ 2 , then c(aij) = n + 2 which is the minimum of the

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labels(colors) used in Case(iii).

Table 2.1.1 Hence, if m > n , χs(Km,n□Km,n) = 2n + 1 for all m, n ≥ 2 .

For m = n , we have the coloring of V (Km,n□Km,n) as as given in the following four cases:

Vertices i j c(aij)

Case(i) 1, 2, 3, ..., m 1, 2, 3, ..., m 1

Case(ii) m + 1, m + 2, m + 3,..., 2m

1, 2, 3, ..., m

 



 





















m j

m i if m

m j

m i if

m j

m i if

,..., 3 , 2 , 1 , 2 , 1 . . .

,..., 3 , 2 , 1 , 2 ,

3 ,..., 3 , 2 , 1 , 1 ,

2 ) c(a

_ij

Case(iii)

1, 2, 3, ..., m m + 1, m + 2, m + 3, ..., 2m

c(aij) =

 



 



















m j m i

if m

m j m i

if m

m j m i

if m

2 , , ...

, 3 , 2 , 1 , 1 2 . . .

2 ,

, ...

, 3 , 2 , 1 , 3

1 ,

, ...

, 3 , 2 , 1 , 2

Case(iv) m +1, m +2, m +3, ..., 2m − 1, 2m

m +1, m +2, m +3, ..., 2m −1, 2m

c(aij)=



















m i for m j

m m

m m i for m j i

2 ,

1 2 ,..., 3 , 2 , 1 ,

1

except for i + j = m(k + 1) + 1 , where k ≥ 2 Also for i = m + 1, m + 2, m + 3, ..., 2 , j = m + 1, m + 2, m + 3, ..., 2m and

If i + j = m(k + 1) + 1 , where k ≥ 2 , we have c(aij) = m + 2, which is the minimum of the labels(colors) used in Case(iii).

Hence, if m = n , χs(Km,n□Km,n) = 2m + 1 for all m, n ≥ 2

For m < n , we have the star coloring of V (Km,n□Km,n) as given in the following four cases:

Vertices aij

i j c(aij)

Case(i)

m + 1, m + 2, m + 3, ..., m

+ n

m + 1, m + 2, m + 3, ..., m + n

1

Case(ii) 1, 2, 3, ..., m m + 1, m + 2, m + 3, ..., m + n

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 





























n m m

m m j i if m

n m m

m m j m i if

n m m

m m j m i if

a c _ij

,..., 3 , 2 , 1 ,

1 , 1 . . .

,..., 3 , 2 , 1 ,

1 ,

3

,..., 3 , 2 , 1 ,

, 2

vertices i j c(aij)

Case(iii)

m + 1, m + 2, m + 3, ..., m +n

1, 2, 3, ..., m

 

 



 





















m j m i

if n

j m i

if m

j m i

if m

a c

_ij

, , 3 , 2 , 1 , 1 2 . . .

2 , , 3 , 2 , 1 , 3

1 , , 3 , 2 , 1 , 2

Case(iv)

1, 2, 3, ..., m 1, 2, 3, ..., m

 



 





















) (mod 1 ,

1 2 . . .

) (mod 4 ,

4 ) (mod 3 ,

3 ) (

m j

i if n

m j

i if m

m j

i if m

a c

_ij

Further for i = 1 and j = 1, 2, 3, ..., m , c(aij) = j and for i = 2, 3, ..., m and j = 1, 2, 3, ..., m

i + j ≡ 2(mod m) , we have c(aij) = i + j .

Thus, if m < n , χs(Km,n□Km,n) = 2m + 1 for all m, n ≥ 2

Hence, in general, χs(Km,n□Km,n) = 2 min {m, n} + 1 , for all m, n ≥ 2 . Illustration 2.1.1: The star coloring of K3,3□K3,3 is given in Fig.2.1

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Fig.2.1 star coloring of K3,3□K3,3

Thus, χs(K3,3□K3,3) = 7 .

2.2

Star Coloring of D(K1,r,K1,s)□K m,n

In this section, the star coloring of the cartesian product of the double star graph D(K1,r,K1,s) with the complete bipartite graph Km,n for given positive integers m, n, r ≥ 2 is discussed.

Theorem 2.2. For given positive integers m, n, r, s ≥ 2 ,

χs(D(K1,r,K1,s)□Km,n) = min{m, n} + 3

Proof. Since Km,n has m + n vertices and D(K1,r, K1,s) has r + s + 2 vertices, G = D(K1,r,K1,s)□Km,n has r + s + 2 layers of m + n vertices each. Let

V1, V2, V3, ..., Vr+s, Vr+s+1, Vr+s+2 denote the r + s + 2 layers of G. Let aij denote a vertex in the i^th layer of the j^th column of V(G).

i.e., V (G) = [a_ij ](r+s+2)×(m+n) .

Let c(aij) denotes the coloring of the vertices aij . The star coloring of vertices aij

is discussed in two cases.

Case(i): m ≥ n

The star coloring of the vertices aij for j = 1, 2, 3, ..., m, i = 1, 2, 3, ..., r, r + 1, r + 2, ..., r + s + 2 is given by

  



 

















m j

s r r

r i if

m j

r i if

m j

r r i

if a

c

_ij

,..., 3 , 2 , 1 , 2 ,...,

4 , 3 ,

3 ,..., 3 , 2 , 1 , 1 ,

2 ,..., 3 , 2 , 1 , 2 , ,..., 3 , 2 , 1 , 1

The star coloring of the vertices aij for j = m + 1, m + 2,...,m + n and i = r, r + 1, r + 2, r + 3 is given by,

 

 



 























) (mod ,

3 } . min{

. . .

) (mod 2 ,

5 ) (mod 1 ,

4 n n r m j i if n m

n r

m j i if

n r

m j i if

a c

_ij

The coloring of the vertices aij for i = 1, 2, 3, ..., r − 1 , j = m + 1, m + 2, ..., m + n follows from the coloring of the vertices aij , for i = r and j = m+1, m+2, ..., m+n .

The coloring of the vertices aij for i = r + 4, r + 5, ..., r + s + 2 , j = m + 1, m + 2, ..., m + n follows from the coloring of the vertices aij , for i = r + 3 and j = m + 1, m + 2, ..., m + n . Thus, from the above discussion it is clear that

χs(D(K1,r,K1,s)□Km,n) = n + 3 Case(ii): m < n

The star coloring of the vertices aij for j = m+1, m+2, ..., m+n and i = 1, 2, 3, ..., r, r+ 1, r + 2, ..., r + s + 2 is given by,

   

 



























n m m

m m j s r r

r i if

n m m

m m j r i if

n m m

m m j r r i

if a

c

_ij

,..., 3 , 2 , 1 ,

2 ,...,

4 , 3 ,

3 ,..., 3 , 2 , 1 ,

1 ,

2 ,..., 3 , 2 , 1 ,

2 ,

,...,

3 ,

2 ,

1 ,

1

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To color j=1,2,…,m and i=r,r+1,r+2,r+3

 



 





















) (mod ,

3 } , min{

. . .

) (mod 2 ,

5 ) (mod 1 ,

4 ) (

m r

j i if n m

m r

j i if

m r

j i if

a c

_ij

The star coloring of the vertices aij for the layers , i = 1, 2, 3, ..., r − 1 , j = 1, 2, ..., m follows from the coloring of the vertices aij of the r^th layer, for j = 1, 2, ..., m .

The coloring of the vertices aij for the layers , i = r + 4, r + 5, ..., r + s + 2 ,

j = 1, 2, ..., m follows from the coloring of the vertices aij of the (r + 3)^th layer, for j = 1, 2, ..., m . Hence, χs(D(K1,r,K1,s)□Km,n) = m + 3 .

Thus, it is evident that, χs(D(K1,r,K1,s)□Km,n) = min{m, n} + 3 for all m,n,r,s ≥ 2.

Conclusion:

In this paper, it is proved that for given positive integers m, n, r, s ≥ 2

(i)

χs(Km,n□Km,n) = 2 min{m, n} + 1 and

(ii)χs(D(K1,r, K1,s)□Km,n) = min{m, n} + 3 . References

[1]

J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, MacMillan, New York (1976) .

[2]

G.Fertin, A.Raspaud, B.Reed, Star Coloring of Graphs, Journal of Graph Theory Vol. 47(3),

163-182, 2004.

[3]

B.Grunbaum, Acyclic colorings of planar graphs, Israel J. Math. Vol.14, 390- 408, 1973.

[4]

R.Hammack, W.Imrich, S.Klavzar, Handbook of Product Graphs, Second Edition, CRC Press, Boca Raton, FL, 2011.

[5]

P.Hemalatha, S.N.Subhathra, Star Coloring of Cartesian Product of Paths and Cycles with Complete Bipartite Graphs, Journal of Computer and Mathematical Sciences. Vol.10(6), 1332-1340, June 2019.

[6]

L.Jethruth Emebla Mary, A.Lydia Mary Juliette Rayen, On Star Coloring of Graphs formed from the Cartesian Product of Some Simple Graphs, International Journal of Mathematics and it’s Applications, Vol.4(4), 115- 122,2016.

[7]

N.Ramya, On Colorings of Wheel Graphs, Indian Journal of Science and Technology, Vol.7(3S), 72-73, 2014.

[8]

M.Subbiah, A Study on Some Variations of Graph Labeling and its Applications in Various Fields, Ph.D thesis submitted to bharadhidasan university , April 2007.

[9]

Tianyong Han, Zehui Shao, En□iang Zhu, Zepeng Li, Fei Deng, Star Coloring of Cartesian Product of Paths and Cycles, Ars Combinatoria, Vol.124, 65-84, 2016.

[10]

Yu-ping Tsao, Chao-Wen Lin, Acyclic chromatic index of Cartesian product of graphs, The 26th Workshop on Combinatorial Mathematics and Computation Theory , 224-232.