Star Coloring of Circulant Graphs

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Star Coloring of Circulant Graphs 𝑮[𝒏; ±{𝟏, 𝟐}]

Jude Annie Cynthia V

¹

and Sindhuja S

^*2

1

Department of Mathematics, Stella Maris College, Chennai-86, INDIA

2

Department of Mathematics, Stella Maris College, Chennai-86, INDIA

email:[email protected], email:[email protected]

(Received on: July 23, 2018) ABSTRACT

Let 𝑮 be an undirected simple graph. Coloring of a graph is an assignment of colors to the vertices, edges or both. A vertex coloring of a graph 𝑮 is said to be proper if no two adjacent vertices have the same color. A Star coloring of a graph 𝑮 is a proper coloring such that no path of length 𝟑 is bicolored. The star chromatic number of a graph 𝑮 is the minimum number of colors which are necessary to star color 𝑮 and it’s denoted by 𝝌_𝒔(𝑮). In this paper, we obtain the exact value of star chromatic number of Circulant graphs 𝑮[𝒏; ±{𝟏, 𝟐}] where 𝒏 ≥ 𝟓 and Mӧbius ladder 𝑴𝒏 where 𝒏 ≥ 𝟒.

Keywords: proper coloring, star coloring, chromatic number, star chromatic number, Circulant graph, Mӧbius ladder.

1. INTRODUCTION

Graph theory is the subject dealing with vertices and edges. It has many applications in various fields. In the past few years, many research works have been done in various topics like labeling, coloring, domination, matching, etc. In graph theory, coloring of a graph 𝐺 = (𝑉, 𝐸) is an assignment of colors to its vertices, edges or both. A vertex coloring of 𝐺 is said to be proper if no two adjacent vertices of 𝐺 have the same color. In the past few decades, several types of vertex coloring have been introduced and still it is one of the active research field in graph theory.

In 1973 the concept of acyclic coloring was introduced by Grunbaum et al.

⁹

and then

he introduced the star coloring as a special type of acyclic coloring and gave the notion of star

chromatic number as 𝜒

_𝑠

(𝐺)

.

According to him the star coloring is the proper coloring on the

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paths with four vertices by giving three distinct colors on it. He proved that any planar graph can be acyclic colored with nine colors, and conjectured that any planar graph can be acyclically colored with five colors. Every star coloring is an acyclic coloring but star coloring a graph typically requires more colors than acyclically coloring the same graph.

The circulant graphs are of particular interest as model of communication network. It has vast number of applications in telecommunication network. They have been used for decades in the design of computer and telecommunication networks due to their optimal fault- tolerance and routing capabilities. The term circulant comes from the nature of its adjacency matrix. A matrix is circulant all its rows are periodic rotations of the first one

⁸

.

Theoretical properties of circulant graphs have been studied extensively and surveyed by Bermond et al.

⁴

. Most of the earlier research concentrated on using circulant graphs to build interconnection networks for distributed and parallel systems. Classes of circulant graphs include cycles, complete graphs, complete bipartite graphs, Paley graphs of prime order, Mӧbius ladder graphs, tetrahedral graph and torus grid graphs.

Mӧbius ladder can also be viewed as a prism with one twisted edge. Mӧbius ladder have many applications in chemistry, chemical stereography, electronics and computer science.

Definition 1.1 A 𝑘-vertex coloring of 𝐺 is an assignment of 𝑘 colors {1, 2…𝑘} to the vertices of 𝐺. The coloring is proper if no two distinct adjacent vertices have the same color

⁴

.

Definition 1.2 If 𝐺 has a proper vertex coloring then the chromatic number of 𝐺 is the minimum number of colors needed to color 𝐺 and is denoted by 𝜒

_𝑠

(𝐺)

⁴

.

Definition 1.3 A star coloring of a graph 𝐺 is a proper coloring such that no path of length 3 in 𝐺 is bicolored

²

.

Definition 1.4 The star chromatic number is the minimum number of colors which are necessary to star color 𝐺 and is denoted by 𝜒

_𝑠

(𝐺)

²

.

Definition 1.5 A Circulant graph denoted by 𝐺(𝑛; ±{1,2. . . 𝑗}), 1 ≤ 𝑗 ≤ ⌊𝑛/2⌋, 𝑛 ≥ 3, is a graph with vertex set 𝑉 = {0,1,2. . . 𝑛 − 1} and the edge set 𝐸 = {(𝑖, 𝑗): |𝑗 − 𝑖| ≡ 𝑠(𝑚𝑜𝑑 𝑛), 𝑠 ∈ {1,2. . . 𝑗}}

⁷

.

Definition 1.6 The Mӧbius ladder is a cubic circulant graph with an even number of vertices and formed from an 𝑛 cycle by adding edges connecting opposite pair of vertices in the cycle except with two pairs which are connected with a twist

¹¹

.

2. LITERATURE SURVEY

A. Vince

¹

introduced the generalization of the chromatic number of a graph such that

the colors are integer modulo 𝑛 and the colors of adjacent vertices are distinct. J.A.Bondy

³

has

given the generalization of the chromatic number of a graph and some simple observation that

lead to a purely combinatorial treatment. Xuding Zhu

¹⁴

has defined an alternate definition of

the star chromatic number, which gives the relation between the star chromatic number and

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the chromatic number; this point of view allows to answer several problems posed by Vince.

Bermond

⁵

has studied the theoretical properties of circulant graphs.

Ko-Wei Lih et al.

⁹

proved that the members of a certain family of circulant graphs are star colorable. Guillaume

²

has given the exact value of star chromatic number of different families of graphs such as trees, cycles, complete bipartite graphs, outer planar graphs and two dimensional grids. R.Arundhadhi, R.Sattanathan

¹³

has given the exact value of star chromatic number of middle, center and total graph of wheel graph families.

R.Arundhadhi, K.Thirusangu

¹²

has given the exact value of star chromatic number of middle, line and total graph of flower graphs. L.Jethruth Emelda Mary, A. Lydia Mary Juliette Rayen

¹⁰

has given that the star chromatic number of some circulant graph. However the star chromatic number in

¹⁰

can still be reduced. J.A.Cynthia et al. have investigated S-(a,d) antimagic labelling

⁶

and Square Difference Labelling

⁸

of circulant graphs, as well as Local Landmarks

⁷

of circulant graphs.

3. STAR CHROMATIC NUMBER OF CIRCULANT GRAPHS

Theorem 3.1: Let 𝐺 be a circulant graph. Then the star chromatic number of 𝐺 is 𝜒

_𝑠

(𝐺)= 5, for 𝑛 ≡ 0, 1(𝑚𝑜𝑑 5)

Proof: Let 𝐺 be a circulant graph with vertex set{𝑣

0

, 𝑣

1

, . . . , 𝑣

𝑛−1

} where 𝑛 ≥ 5. By the definition of circulant graph, each vertex 𝑣

_𝑖

is adjacent to {𝑣

_𝑖−2

, 𝑣

_𝑖−1

, 𝑣

_𝑖+1

, 𝑣

_𝑖+2

} vertices. Let us assign star colors to the vertices of the graph.

Let 𝑓

₁

:𝑉(𝐺) → {0, 1, 2, 3, 4}. Then 𝑓

₁

is defined as follows:

Case 1: If 𝑛 ≡ 0 (𝑚𝑜𝑑 5). Then 𝑓

1

(𝑣

i

) = {

(𝑖 + 1)(𝑚𝑜𝑑 5), 0 ≤ 𝑖 ≤ 𝑛 − 6 0, 𝑖 = 𝑛 − 1

1, 𝑖 = 𝑛 − 5 𝑗, 𝑖 = 𝑛 − 𝑗 , 𝑗 = 2, 3, 4 Case 2: If 𝑛 ≡ 1 (𝑚𝑜𝑑 5). Then

𝑓

₁

(𝑣

i

) = { (𝑖 + 1)(𝑚𝑜𝑑 5), 0 ≤ 𝑖 ≤ 𝑛 − 2 3 , 𝑖 = 𝑛 − 1

There are different paths on vertices and the assignment of colors is as follows:



𝑣

_𝑖

, 𝑣

_𝑖+1

, 𝑣

_𝑖+2

, 𝑣

_𝑖+3

with colors 1,2,3,4



𝑣

_𝑖

, 𝑣

_𝑖+2

, 𝑣

_𝑖+4

, 𝑣

_𝑖+6

with colors 1,3,0,2



𝑣

_𝑖

, 𝑣

_𝑖+1

, 𝑣

_𝑖+3

, 𝑣

_𝑖+5

with colors 1,2,4,1



𝑣

_𝑖

, 𝑣

_𝑖+1

, 𝑣

_𝑖+2

, 𝑣

_𝑖+4

with colors 1,2,3,0



𝑣

_𝑖

, 𝑣

_𝑖+2

, 𝑣

_𝑖+3

, 𝑣

_𝑖+4

with colors 1,3,4,0



𝑣

_𝑖

, 𝑣

_𝑖+2

, 𝑣

_𝑖+4

, 𝑣

_𝑖+5

with colors 1,3,0,1

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Figure 3.1: Circulant graph 𝑮[𝟏𝟔; ±{𝟏, 𝟐}]

Thus we can see that 𝐺[𝑛; ±{1,2}] has star chromatic number to be 5, for 𝑛 ≡ 0, 1(𝑚𝑜𝑑 5).

Theorem 3.2: Let 𝐺 be a circulant graph. Then the star chromatic number of 𝐺 is 𝜒

s

(𝐺) = 6, for 𝑛 ≡ 2,3,4 (𝑚𝑜𝑑 5)

Proof: We further classify the above values of 𝑛 into 6 subcases.

Let 𝑓

₂

: 𝑉(𝐺) → {0, 1, 2, 3, 4, 5}. Then 𝑓

₂

is defined as follows:

Sub case 1.1: If 𝑛 ≡ 0, 5 (𝑚𝑜𝑑 6). Then 𝑓

₂

(𝑣

i

) = (𝑖 + 1)(𝑚𝑜𝑑 6), 0 ≤ 𝑖 ≤ 𝑛 − 1 Sub case 1.2: If 𝑛 ≡ 2 (𝑚𝑜𝑑 6). Then 𝑓

₂

(𝑣

i

) = {

(𝑖 + 1)(𝑚𝑜𝑑 6), 0 ≤ 𝑖 ≤ 𝑛 − 3 3 , 𝑖 = 𝑛 − 2

4 , 𝑖 = 𝑛 − 1

Sub case 1.3: If 𝑛 ≡ 4 (𝑚𝑜𝑑 6). Then

𝑓

₂

(𝑣

i

) = {

(𝑖 + 1)(𝑚𝑜𝑑 6), 0 ≤ 𝑖 ≤ 𝑛 − 5 2, 𝑖 = 𝑛 − 4

3, 𝑖 = 𝑛 − 3 4, 𝑖 = 𝑛 − 2 0, 𝑖 = 𝑛 − 1 Sub case 1.4: If 𝑛 ≡ 1 (𝑚𝑜𝑑 6). Then 𝑓

₂

(𝑣

i

) = { (𝑖 + 1)(𝑚𝑜𝑑 6), 0 ≤ 𝑖 ≤ 𝑛 − 2

4 , 𝑖 = 𝑛 − 1

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Sub case 1.5: If 𝑛 ≡ 3 (𝑚𝑜𝑑 6). Then

𝑓

₂

(𝑣

i

) = {

(𝑖 + 1)(𝑚𝑜𝑑 6), 0 ≤ 𝑖 ≤ 𝑛 − 5 3, 𝑖 = 𝑛 − 4

1, 𝑖 = 𝑛 − 3 0, 𝑖 = 𝑛 − 2 4, 𝑖 = 𝑛 − 1

Figure 3.2: Circulant graph 𝑮[𝟐𝟐; ±{𝟏, 𝟐}]

There are 6 different paths on vertices and the assignment of colors is as follows:



𝑣

_𝑖

, 𝑣

_𝑖+1

, 𝑣

_𝑖+2

, 𝑣

_𝑖+3

with colors 1,2,3,4



𝑣

_𝑖

, 𝑣

_𝑖+2

, 𝑣

_𝑖+4

, 𝑣

_𝑖+6

with colors 1,3,5,1



𝑣

𝑖

, 𝑣

𝑖+1

, 𝑣

𝑖+3

, 𝑣

𝑖+5

with colors 1,2,4,0



𝑣

_𝑖

, 𝑣

_𝑖+1

, 𝑣

_𝑖+2

, 𝑣

_𝑖+4

with colors 1,2,3,5



𝑣

_𝑖

, 𝑣

_𝑖+2

, 𝑣

_𝑖+3

, 𝑣

_𝑖+4

with colors 1,3,4,5



𝑣

_𝑖

, 𝑣

_𝑖+2

, 𝑣

_𝑖+4

, 𝑣

_𝑖+5

with colors 1,3,4,0

We can check that the above functions give a star coloring to the circulant graph. Thus we have exact value of star chromatic number of circulant graphs 𝐺[𝑛; ±{1,2}] to be 6 when 𝑛 ≥ 5, 𝑛 ≡ 2, 3, 4 (𝑚𝑜𝑑 5).

Theorem 3.3: Let 𝑀

n

be a Mӧbius ladder. Then the star chromatic number of 𝑀

n

is 𝜒

_𝑠

(𝑀

n

) = { 4, 𝑛 = 4

5, 𝑛 ≥ 5

Proof: Let 𝑀

n

be a Mӧbius ladder with 2𝑛 vertices has vertex set 𝑢

i

and 𝑣

i

where 1 ≤ 𝑖 ≤ 𝑛.

The adjacencies of the vertices are as follows:



𝑢

i

is adjacent to 𝑣

i,

1 ≤ 𝑖 ≤ 𝑛



𝑢

i

is adjacent to 𝑢

i+1,

1 ≤ 𝑖 ≤ 𝑛 − 1

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

𝑣

i

is adjacent to 𝑣

i+1,

1 ≤ 𝑖 ≤ 𝑛 − 1



𝑢

₁

is adjacent to 𝑣

n



𝑢

n

is adjacent to 𝑣

₁

(Note that the name comes from a “twist” in one side) Let us assign star color to the vertices of the Mӧbius ladder.

Let 𝑓

₁ :

𝑢

i

(𝑀

n

) → {1,2,3,4,5}. Then 𝑓

₁

is defined as follows:

𝑓

₁

(𝑢

i

)=𝑖, 1 ≤ 𝑖 ≤ 3 𝑓

₁

(𝑢

i

) ={

4, 𝑖 ≡ 0(𝑚𝑜𝑑 4) 5, 𝑖 ≡ 1,3(𝑚𝑜𝑑 4)

2, 𝑖 ≡ 2(𝑚𝑜𝑑 4)

, 4 ≤ 𝑖 ≤ 𝑛

Let 𝑓

₂

: 𝑣

i

(𝑀

n

) → {1,2,3,4}. Then 𝑓

₂

is defined as follows:

𝑓

₂

(𝑣

i

) = {

3, 𝑖 = 1 4, 𝑖 = 2 1, 𝑖 = 3 𝑓

₂

(𝑣

i

) ={

2, 𝑖 ≡ 0(𝑚𝑜𝑑 4) 3, 𝑖 ≡ 1,3(𝑚𝑜𝑑 4)

4, 𝑖 ≡ 4(𝑚𝑜𝑑 4)

, 4 ≤ 𝑖 ≤ 𝑛

Figure 3.3: Mӧbius ladder 𝑴6

There are 9 different paths on 4 vertices and the assignment of colors is as follows:



𝑢

i

, 𝑢

i+1

, 𝑢

i+2

, 𝑢

i+3

with colors 1,2,3,4



𝑣

i

, 𝑣

i+1

, 𝑣

i+2

, 𝑣

i+3

with colors 3,4,1,2



𝑣

i

, 𝑢

i

, 𝑢

i+1

, 𝑣

i+1

with colors 3,1,2,4



𝑣

i

, 𝑢

i

, 𝑢

i+1

, 𝑢

i+2

with colors 3,1,2,3



𝑣

i

, 𝑣

i+1

, 𝑢

i+1

, 𝑢

i+2

, with colors 3.4.2,3



𝑣

i

, 𝑢

i

, 𝑢

n

, 𝑣

n

with colors 3,1,2,4



𝑣

i

, 𝑢

i

, 𝑣

n

, 𝑢

n

with colors 3,1,4,2



𝑣

i

, 𝑢

n

, 𝑣

n

, 𝑢

i

with colors 3,2,4,1



𝑣

n

, 𝑢

i

, 𝑢

n

, 𝑣

i

with colors 4,1,2,3.

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Thus we have the exact value of star chromatic number of Mӧbius ladder 𝑀

_𝑛

to be 4, for 𝑛 = 4 and 5, for 𝑛 ≥ 5.

5. CONCLUSION

In this paper we have determined the exact value of star chromatic number of circulant graph 𝐺[𝑛; ±{1,2}], where 𝑛 ≥ 5 and Mӧbius ladder 𝑀

_𝑛

, where 𝑛 ≥ 4. Further, the star chromatic numbers of cycle and wheel related graphs are under study.

REFERENCES

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(1976).

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