Cycle Related Balanced Divided Square Difference Cordial Graphs

Cycle Related Balanced Divided Square Difference Cordial Graphs

A. Alfred Leo* and R. Vikramaprasad Research Scholar,

Research and Development Centre, Bharathiar University, Coimbatore-641046,

Tamil Nadu, INDIA.

email:[email protected] Assistant Professor, Department of Mathematics, Government Arts College, Salem-636007,

Tamil Nadu, INDIA.

(Received on: June 15, 2018)

ABSTRACT

In this article, the balanced divided square difference cordial behavior of some cycle related graphs such as cycle graph C_n with one chord, sun flower graph, gear graph, double wheel graph, G^′= 〈Wn1, Wn2〉, dragon graph were discussed.

Keywords: balanced cordial, cycle graph 𝐶𝑛 with one chord, sun flower graph, gear graph, double wheel graph, 𝐺^′ = 〈𝑊_𝑛¹, 𝑊_𝑛²〉, dragon graph.

1. INTRODUCTION

The field of Graph Theory plays an important role in various areas of pure and applied sciences. By a graph, we mean a finite, undirected graph without loops and multiple edges.

For terms and notations we refer to Harary

. In 1967, Rosa

introduced a labeling of G called β-valuation. For all detailed survey of graph labeling we refer to Gallian

. Cahit

have introduced the concept of cordial labeling. R.Varatharajan, et al.

introduced the notion of divisor cordial labeling. Also Varatharajan, et al.

introduced the special classes of divisor cordial graphs. Dhavaseelan et al.

introduced the concept of even sum cordial labeling graphs.

V. J. Kaneria et al.

have introduced the concept of balanced cordial labeling. The concept of

divided square difference cordial graphs was introduced by A. Alfred Leo et al.

. Also

A. Alfred Leo et al.

have discussed some more divided square difference cordial graphs.

In this paper, we have discussed the balanced divided square difference cordial behavior of some cycle related graphs called cycle graph 𝐶

with one chord, sun flower graph, gear graph, double wheel graph, 𝐺

= 〈𝑊

, 𝑊

〉, dragon graph.

2. PRELIMINARIES

Definition 2.1 [6]

A vertex labeling of a graph G is an assignment f of labels to the vertices of G that induces each edge uv a label depending on the vertex label f(u) and f(v).

Definition 2.2 [6]

A mapping 𝑓: 𝑉(𝐺) → {0,1} is called binary vertex labeling of G and 𝑓(𝑉) is called the label of the vertex 𝑣 of G under 𝑓.

Definition 2.3 [3]

A binary vertex labeling f of a graph G is called a cordial labeling if |𝑣

(0) − 𝑣

(1)| ≤ 1 and

|𝑒

(0) − 𝑒

(1)| ≤ 1.

A graph G is cordial if it admits cordial labeling.

Definition 2.4

A chord of cycle 𝐶

is an edge joining two non-adjacent vertices of cycle 𝐶

. Definition 2.5 [9]

A flower graph 𝐹𝑙

is the graph obtained from the Helm graph 𝐻

by joining each pendent vertex to apex of the Helm 𝐻

. For this graph |𝑉(𝐺)| = 2𝑛 + 1 and |𝐸(𝐺)| = 4𝑛.

Definition 2.6 [10]

A sun flower graph 𝑆𝐹𝑙

is the graph obtained from the flower graph by adding n pendent edges to the central vertex. For this graph |𝑉(𝐺)| = 3𝑛 + 1 and |𝐸(𝐺)| = 5𝑛

Definition 2.7 [4]

A gear graph 𝐺

is obtained from the wheel graph 𝑊

by adding a vertex between every pair of adjacent vertices of the cycle.

Definition 2.8 [4]

The double wheel graph 𝐷𝑊

of size n can be composed of 2𝐶

+ 𝐾

. ie) it consists of two cycles of size n where the vertices of two cycles are all connected to a common hub.

Definition 2.9 [9]

A dragon graph 𝐷

is obtained by joining a vertex of a cycle 𝐶

with a pendent vertex of a path 𝑃

.

Definition 2.10 [8]

A cordial graph G with a cordial labeling 𝑓 is called a balanced cordial graph if

|𝑒

(0) − 𝑒

(1)| = |𝑣

(0) − 𝑣

(1)| = 0.

It is said to be edge balanced cordial graph if |𝑒

(0) − 𝑒

(1)| = 0 and |𝑣

(0) − 𝑣

(1)| = 1.

Similarly it is said to be vertex balanced cordial graph if |𝑒

(0) − 𝑒

(1)| = 1 and

|𝑣

(0) − 𝑣

(1)| = 0.

A cordial graph G is said to be unbalanced cordial graph if

|𝑒

(0) − 𝑒

(1)| = |𝑣

(0) − 𝑣

(1)| = 1.

Definition 2.11 [1]

Let 𝐺 = (𝑉, 𝐸) be a simple graph and 𝑓: 𝑉 → {1,2,3, … |𝑉|} be bijection. For each edge 𝑢𝑣, assign the label 1 if |

𝑓(𝑢)−𝑓(𝑣)

| is odd and the label 0 otherwise. f is called divided square difference cordial labeling if |𝑒

(0) − 𝑒

(1)| ≤ 1, where 𝑒

(1) and 𝑒

(0) denote the number of edges labeled with 1 and not labeled with 1 respectively.

A graph G is called divided square difference cordial if it admits divided square difference cordial labeling.

Definition 2.12

A divided square difference cordial graph G is called a balanced divided square difference cordial graph if |𝑒

(0) − 𝑒

(1)| = 0.

A divided square difference cordial graph G is called a unbalanced divided square difference cordial graph if |𝑒

(0) − 𝑒

(1)| = 1.

Proposition 2.13 [1]

1. Any path 𝑃

is a divided square difference cordial graph.

2. Any cycle 𝐶

is a divided square difference cordial graph except 𝑛 = 6, 6 + 𝑑, 6 + 2𝑑, … when 𝑑 = 4.

Proposition 2.14 [2]

1. The wheel graph 𝑊

is a divided square difference cordial graph for 𝑛 ≡ 0,1 (𝑚𝑜𝑑 4).

2. The flower graph 𝐹𝑙

is a divided square difference cordial for 𝑛 ≡ 0,1 (𝑚𝑜𝑑 4).

Note 2.15

In this article, we have consider the cycle graph 𝐶

except for 𝑛 ≡ 2 (𝑚𝑜𝑑 4).

3. MAIN RESULT Proposition 3.1

The cycle graph 𝐶

with one chord is a balanced divided square difference cordial when 𝑛 is odd.

Proof

Let G be a cycle graph 𝐶

with one chord having |𝑉(𝐺)| = 𝑛 and |𝐸(𝐺)| = 𝑛 + 1.

Let 𝑣

, 𝑣

, … , 𝑣

are the vertices of the cycle graph 𝐶

. Then Join any two non-adjacent

vertices of cycle 𝐶

with a chord. Define a map 𝑓: 𝑉(𝐺) → {1,2, … , 𝑛}. Label the vertices of

cycle 𝐶

by Proposition 2.13.

Case i: When 𝑛 ≡ 0(𝑚𝑜𝑑 4)

Join any two non-adjacent vertices of cycle 𝐶

with a chord.

Thus we get, |𝑒

(0) − 𝑒

(1)| ≤ 1.

Case ii: When 𝑛 ≡ 2(𝑚𝑜𝑑 4)

Join any two non-adjacent vertices (both having even or odd label) of cycle 𝐶

with a chord.

Thus we get, |𝑒

(0) − 𝑒

(1)| ≤ 1.

Case iii: When n is odd

Join two non adjacent vertices (one with odd label and one with even label) of cycle 𝐶

with a chord.

Thus, we get |𝑒

(0) − 𝑒

(1)| ≤ 1.

In particular, we get |𝑒

(0) − 𝑒

(1)| = 0.

Hence G is a balanced divided square difference cordial graph when 𝑛 is odd and unbalanced cordial when 𝑛 is even.

Example 3.2

Fig 1. 𝑪𝟖 with one chord Fig 2. 𝑪𝟗 with one chord

Proposition 3.3

The sunflower graph 𝑆𝐹𝑙

is a balanced divided square difference cordial when 𝑛 ≡ 0 (𝑚𝑜𝑑 4).

Proof

Let G be a sunflower graph 𝑆𝐹𝑙

with |𝑉(𝐺)| = 3𝑛 + 1 and |𝐸(𝐺)| = 5𝑛. First we can

construct and label the flower graph 𝐹𝑙

by proposition 2.14. Then add n pendent edges to the

central vertex of 𝐹𝑙

and assign the vertices as 𝑤

, 𝑤

, … , 𝑤

.

Label the vertices by taking (𝑤

) = 2𝑛 + 𝑖 + 1, 1 ≤ 𝑖 ≤ 𝑛 . Thus, we get |𝑒

(0) − 𝑒

(1)| ≤ 1.

In particular, |𝑒

(0) − 𝑒

(1)| = 0 when 𝑛 ≡ 0 (𝑚𝑜𝑑 4) and |𝑒

(0) − 𝑒

(1)| = 1 when 𝑛 ≡ 1 (𝑚𝑜𝑑 4).

Hence G is a balanced divided square difference cordial graph when 𝑛 ≡ 0 (𝑚𝑜𝑑 4) and unbalanced divided square difference cordial when 𝑛 ≡ 1 (𝑚𝑜𝑑 4).

Example 3.4

Fig 3. Sunflower graph 𝑺𝑭𝒍_𝟖

Proposition 3.5

The gear graph 𝐺

is a balanced divided square difference cordial when 𝑛 is even.

Proof

Let G be a gear graph 𝐺

with |𝑉(𝐺)| = 2𝑛 + 1 and |𝐸(𝐺)| = 3𝑛. Let 𝑣

, 𝑣

, … , 𝑣

are the vertices of the cycle 𝐶

and x be the apex vertex. First we draw the wheel graph 𝑊

by Proposition 2.14. Then add a vertex between every pair of adjacent vertices of the cycle 𝐶

. We assign the label for the vertices as 𝑢

, 𝑢

, … , 𝑢

. Label the new vertices 𝑢

, 𝑢

, … , 𝑢

by taking 𝑓(𝑢

) = 𝑛 + 𝑖 + 1, 1 ≤ 𝑖 ≤ 𝑛.

Thus, we get |𝑒

(0) − 𝑒

(1)| ≤ 1.

In particular, |𝑒

(0) − 𝑒

(1)| = 0 when 𝑛 is even and |𝑒

(0) − 𝑒

(1)| = 1 when 𝑛 is odd.

Hence G is a balanced divided square difference cordial graph when 𝑛 is even and unbalanced

divided square difference cordial when 𝑛 is odd.

Example 3.6

Fig 4. Gear graph 𝑮_𝟏𝟐

Proposition 3.7

The double wheel graph 𝐷𝑊

, (𝑛 ≡ 0,1 (𝑚𝑜𝑑 4)) is a balanced divided square difference cordial.

Proof

Let G be a double wheel graph 𝐷𝑊

, (𝑛 ≡ 0,1 (𝑚𝑜𝑑 4)) with |𝑉(𝐺)| = 2𝑛 + 1 and

|𝐸(𝐺)| = 4𝑛.

Consider two cycles, inner cycle 𝐶

and outer cycle 𝐶

. Let 𝑣

, 𝑣

, … , 𝑣

are the vertices of cycle 𝐶

and 𝑢 is the apex vertex. Then draw the wheel graph 𝑊

by Proposition 2.14. Then introduce the second cycle graph 𝐶

with vertices 𝑢

, 𝑢

, … , 𝑢

and join the vertices of 𝐶

with the apex vertex 𝑢 which is common for both 𝑊

and 𝑊

. We can now label the cycle graph 𝐶

using proposition 2.14 by taking 𝑓(𝑢

) = 𝑛 + 2.

Then we get |𝑒

(0) − 𝑒

(1)| ≤ 1. In particular, |𝑒

(0) − 𝑒

(1)| = 0.

Hence G is a balanced divided square difference cordial graph.

Example 3.8

Fig 5. Double wheel graph 𝑫𝑾_𝟖

Proposition 3.9

The graph 𝐺

= 〈𝑊

, 𝑊

〉, (𝑛 is even) is a balanced divided square difference cordial.

Proof

Let G be a 𝐺

= 〈𝑊

, 𝑊

〉, (𝑛 is even) graph with |𝑉(𝐺)| = 2𝑛 + 3, |𝐸(𝐺)| = 4𝑛 + 2. Let 𝑢

, 𝑢

, … , 𝑢

are the vertices of 𝑊

and 𝑣

, 𝑣

, … , 𝑣

are the vertices of 𝑊

. Let 𝑥, 𝑦 are the apex vertices of 𝑊

, 𝑊

respectively. Let 𝑤 be the vertex joining to apex vertices of 𝑊

and 𝑊

.

Case i: when 𝑛 ≡ 0(𝑚𝑜𝑑 4)

First construct the wheel graphs 𝑊

and 𝑊

using Proposition 2.14 by taking 𝑓(𝑢

) = 1 and (𝑣

) = 𝑛 + 2 .

Then connect the apex vertices 𝑥, 𝑦 to 𝑤 and label the vertex 𝑤 by 𝑓(𝑤) = 2𝑛 + 3.

Then we get |𝑒

(0) − 𝑒

(1)| ≤ 1.

Case ii: when 𝑛 ≡ 2(𝑚𝑜𝑑 4)

First construct the wheel graphs 𝑊

and 𝑊

using Proposition 2.14 by taking 𝑓(𝑢

) = 1, … , 𝑓(𝑢

) = 𝑛 − 1, 𝑓(𝑢

) = 𝑛 and

𝑓(𝑣

) = 𝑛 + 2, … , 𝑓(𝑣

) = 2𝑛 + 1, 𝑓(𝑣

) = 2𝑛.

Then connect the apex vertices 𝑥, 𝑦 to 𝑤 and label the vertex 𝑤 by 𝑓(𝑤) = 2𝑛 + 3.

Thus, we get |𝑒

(0) − 𝑒

(1)| ≤ 1.

In particular, |𝑒

(0) − 𝑒

(1)| = 0 in both the cases.

Hence G is a balanced divided square difference cordial graph when 𝑛 is even.

Note:

When 𝑛 is odd, 𝐺 is not a divided square difference cordial graph.

Example 3.10

Fig 7. 𝑮^′= 〈𝑾_𝟖^𝟏, 𝑾_𝟖^𝟐〉

Proposition 3.11

The Dragon graph 𝐷

(except 𝑛 ≡ 2 (𝑚𝑜𝑑4)) is a balanced divided square difference cordial when both 𝑚, 𝑛 are odd or both are even.

Proof

Let G be a Dragon graph 𝐷

(except 𝑛 ≡ 2 (𝑚𝑜𝑑4)) with |𝑉(𝐺)| = 𝑚 + 𝑛 = |𝐸(𝐺)|. Let 𝑃

: 𝑣

, 𝑣

, … , 𝑣

be a path and 𝐶

: 𝑢

, 𝑢

, … , 𝑢

be a cycle.

Let the dragon be obtained by making 𝑣

and 𝑢

as adjacent. Then the vertex set of dragon D is {𝑣

, 𝑣

, … , 𝑣

, 𝑢

, 𝑢

, … , 𝑢

}. Now, define a map 𝑓: 𝑉(𝐺) → {1,2, … , 𝑚 + 𝑛} as follows.

First, we can construct the path 𝑃

by Proposition 2.13. Then construct the cycle 𝐶

by Proposition 2.13 taking 𝑓(𝑢

) = 𝑚 + 1. Now join 𝑣

and 𝑢

to get the dragon graph D.

Case i: 𝑚 is even and 𝑛 is even In this case |𝑒

(0) − 𝑒

(1)| = 0.

Case ii: 𝑚 is odd and 𝑛 is odd In this case |𝑒

(0) − 𝑒

(1)| = 0.

Case iii: 𝑚 is odd and 𝑛 is even In this case |𝑒

(0) − 𝑒

(1)| = 1.

Case iv: 𝑚 is even and 𝑛 is odd In this case |𝑒

(0) − 𝑒

(1)| = 1.

Therefore in general, we get |𝑒

(0) − 𝑒

(1)| ≤ 1.

Hence G is a balanced divided square difference cordial graph when both 𝑚, 𝑛 are odd or both

are even. Otherwise G is unbalanced.

Note: For 𝑛 ≡ 2 (𝑚𝑜𝑑 4), Dragon graph 𝐷

is not a divided square difference cordial graph.

Example 3.12

Fig 8. Dragon graph D

4. CONCLUSION

In this paper, the balanced divided square difference cordial labeling behavior of some cycle related graphs such as cycle graph 𝐶

with one chord, sun flower graph, gear graph, double wheel graph, 𝐺

= 〈𝑊

, 𝑊

〉, dragon graph were discussed.

ACKNOWLEDGMENT

The authors are highly thankful to the anonymous referees for constructive suggestions and comments. Also thankful to Dr. R. Dhavaseelan for his continuous support, feedback and comments.

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