Vol 7, No 3 (2016)

Available online throug

ISSN 2229 – 5046

SPLITTING GRAPH ON EVEN SUM CORDIAL LABELING OF GRAPHS

𝟐𝟐

_{R. VIKRAMAPRASAD,}

_{R. DHAVASEELAN,}

_{S. ABHIRAMI}

𝟏𝟏,𝟑𝟑

_{Department of Mathematics,}

Sona College of Technology, Salem - 636 005, Tamil Nadu, India.

𝟐𝟐

_{Department of Mathematics,}

A.A Government Arts College, Musiri-621 201, Tamil Nadu, India.

(Received On: 02-03-16; Revised & Accepted On: 28-03-16)

ABSTRACT

I

Keywords: Splitting graphs, Helm graphs, Flower graphs, Bistar.

2010 Mathematics Subject Classification: 05C78.

1. INTRODUCTION

The origin of graph labeling can be attributed to Rosa[7].In [1],Cahit introduce the concept of cordial labeling of graph. For splitting graph we refer E.Sampath kumar and H.B.Waliker[8]. For different splitting graphs we refer R.Lawrence Razario raj[5, 6]. In [9, 10], Vaidya, et.al., introduced the concepts of Helm, Flower and Bistar graphs are divisor cordial labeling. In this paper, we investigate the splitting graph of the family of bipartite graphs, paths and cycles are even sum cordial graphs and proved several classes of graphs such that 𝑃𝑃_𝑚𝑚(+)𝐾𝐾_𝑛𝑛,(𝐾𝐾_𝑛𝑛∪ 𝑃𝑃_𝑚𝑚) + 2𝐾𝐾₁,〈𝑊𝑊_𝑛𝑛(1),𝑊𝑊_𝑛𝑛(2)〉,

𝐵𝐵𝑛𝑛,𝑛𝑛,𝑆𝑆(𝐵𝐵𝑛𝑛,𝑛𝑛), Helm graph 𝐻𝐻𝑛𝑛 and Flower graph 𝐹𝐹𝑙𝑙𝑛𝑛 are even sum cordial graphs.

2. PRELIMINARIES

Definition 2.1 [3] Let 𝐺𝐺= (𝑉𝑉,𝐸𝐸) be a simple graph and 𝑓𝑓:𝑉𝑉 →{1,2,3 … . |𝑉𝑉|} be a bijection. For each edge uv, assign

the label 1 if 𝑓𝑓(𝑢𝑢) +𝑓𝑓(𝑣𝑣) is even and the label 0 otherwise. f is called an even sum cordial labeling if

�𝑒𝑒𝑓𝑓(0)− 𝑒𝑒𝑓𝑓(1)� ≤1, where 𝑒𝑒𝑓𝑓(1) and 𝑒𝑒𝑓𝑓(0)denote the number of edges labeled with 1 and number of edges labeled

with 0 respectively. A graph with an even sum cordial labeling is called an even sum cordial graph.

Proposition 2.1: [3] Any path is an even sum cordial graph.

Proposition 2.2: [3] Any cycle 𝐶𝐶_𝑛𝑛 is an even sum cordial graph except 𝑛𝑛= 6,6 +𝑑𝑑, 6 + 2𝑑𝑑, . .. when 𝑑𝑑= 4 .

Definition 2.2: [6] A wheel graph 𝑊𝑊_𝑛𝑛 is a graph with 𝑛𝑛+ 1 vertices formed by connecting a single vertex to all the

vertices of 𝑛𝑛 cycle.

Definition 2.3: [9] The Helm graph 𝐻𝐻_𝑛𝑛 is the graph obtained from a wheel 𝑊𝑊_𝑛𝑛 by attaching a pendent edge to each rim

vertex. It consists three types of vertices: • an apex of degree n

• n vertices of degree 4 • n pendent vertices.

Corresponding Author:

_{S. Abhirami,}

_{Department of Mathematics,}

Definition 2.4: [9] The Flower graph 𝐹𝐹𝑙𝑙_𝑛𝑛 is the graph obtained from Helm 𝐻𝐻_𝑛𝑛 by joining each pendent vertex to apex of the Helm 𝐻𝐻_𝑛𝑛 . It consists three types of vertices:

• an apex of degree 2n • n vertices of degree 4 • n vertices of degree 2.

Definition 2.5: [6] Consider two copies of graph 𝐺𝐺 namely 𝐺𝐺₁ and 𝐺𝐺₂.Then the graph 𝐺𝐺′=〈𝐺𝐺₁,𝐺𝐺₂〉 is a graph obtained

by joining the apex vertices of 𝐺𝐺₁ and 𝐺𝐺₂ by a new vertex.

Definition 2.6: [10] The Bistar 𝐵𝐵_{𝑛𝑛,𝑛𝑛} is a graph obtained by joining the two copies of 𝐾𝐾_1,𝑛𝑛 by an edge is called bistar

graph.

Definition 2.7: [5] The graph 𝑃𝑃_𝑚𝑚(+)𝐾𝐾_𝑛𝑛 is a graph with the vertex set 𝑉𝑉(𝐺𝐺) = {𝑢𝑢_𝑖𝑖,𝑣𝑣_𝑗𝑗, 1≤ 𝑖𝑖 ≤ 𝑚𝑚, 1≤ 𝑗𝑗 ≤ 𝑛𝑛} and the

edge set 𝐸𝐸(𝐺𝐺) = {𝑢𝑢_𝑖𝑖𝑢𝑢_𝑖𝑖+1,𝑢𝑢₁𝑣𝑣_𝑗𝑗,𝑢𝑢_𝑚𝑚𝑣𝑣_𝑗𝑗, 1≤ 𝑖𝑖 ≤(𝑚𝑚 −1),1≤ 𝑗𝑗 ≤ 𝑛𝑛}

Definition 2.8: [4] The union of two graphs 𝐺𝐺₁= (𝑉𝑉₁,𝐸𝐸₁) and 𝐺𝐺₂= (𝑉𝑉₂,𝐸𝐸₂) is another graph 𝐺𝐺₃ ie.,𝐺𝐺₃=𝐺𝐺₁∪ 𝐺𝐺₂

whose vertex set 𝑉𝑉₃=𝑉𝑉₁∪ 𝑉𝑉₂ and the edge set is 𝐸𝐸₃=𝐸𝐸₁∪ 𝐸𝐸₂.

Definition 2.9: [4] The joint 𝐺𝐺₁+𝐺𝐺₂ of 𝐺𝐺₁ and 𝐺𝐺₂ consists of 𝐺𝐺₁∪ 𝐺𝐺₂ and all lines joining 𝑉𝑉₁ with 𝑉𝑉₂.

Definition 2.10: [8] (Splitting Graph) For each vertex 𝑣𝑣 of a graph 𝐺𝐺, take a new vertex 𝑣𝑣′, join 𝑣𝑣′ to all vertices of G

adjacent to 𝑣𝑣. The graph 𝑆𝑆(𝐺𝐺) thus obtained is called splitting graph of 𝐺𝐺.

3. MAIN RESULTS

Proposition 3.1: The graph 𝑆𝑆(𝐾𝐾_1,𝑚𝑚) is an even sum cordial graph.

Example 3.1: 𝑺𝑺(𝑲𝑲_{𝟏𝟏,𝟑𝟑})

Proposition 3.2: The graph 𝑆𝑆(𝐾𝐾_2,𝑚𝑚) is an even sum cordial graph.

Example 3.2: 𝑺𝑺(𝑲𝑲_{𝟐𝟐,𝟑𝟑})

Proposition 3.3: The graph 𝑆𝑆(𝐾𝐾_{1,𝑛𝑛,𝑛𝑛}) is an even sum cordial graph.

Proposition 3.4: The graph 𝑆𝑆(𝑃𝑃_𝑛𝑛) is an even sum cordial graph except n is multiple of 11.

Proof: Let G be a graph 𝑆𝑆(𝑃𝑃_𝑛𝑛) Let 𝑣𝑣₁,𝑣𝑣₂, . . . ,𝑣𝑣_𝑛𝑛 be the vertices of 𝑃𝑃_𝑛𝑛.By Proposition 2.1. we construct the even sum

cordial path 𝑃𝑃_𝑛𝑛. Then add the vertices 𝑣𝑣₁′,𝑣𝑣₂′, . . . ,𝑣𝑣_𝑛𝑛′ to the graph 𝑃𝑃_𝑛𝑛.Assign the labeling for the vertices in the following way. Define 𝑓𝑓:𝑉𝑉(𝐺𝐺)→{1,2,3, . . . ,2𝑛𝑛} as follows 𝑓𝑓�𝑣𝑣_{𝑛𝑛−𝑖𝑖}′ �=𝑛𝑛+𝑖𝑖+ 1,0≤ 𝑖𝑖 ≤ 𝑛𝑛 −1 and |𝑉𝑉(𝐺𝐺)| = 2𝑛𝑛, |𝐸𝐸(𝐺𝐺)| =

3(𝑛𝑛 −1). Now, we construct the splitting graph 𝑆𝑆(𝑃𝑃_𝑛𝑛) by definition 2.10. Thus |𝑒𝑒_𝑓𝑓(0)− 𝑒𝑒_𝑓𝑓(1)|≤1. Hence G is an even sum cordial graph.

Example 3.4: 𝑺𝑺(𝑷𝑷_{𝟏𝟏𝟐𝟐})

Proposition 3.5: The graph 𝑆𝑆(𝐶𝐶_𝑛𝑛) is an even sum cordial graph except 𝑛𝑛= 6, 6 +𝑑𝑑, 6 + 2𝑑𝑑, when 𝑑𝑑= 4.

Proof. Let G be a graph 𝑆𝑆(𝐶𝐶_𝑛𝑛) Let 𝑣𝑣₁,𝑣𝑣₂, . . . ,𝑣𝑣_𝑛𝑛 be the vertices of 𝐶𝐶_𝑛𝑛. By Proposition 2.2. we construct the even sum cordial cycle 𝐶𝐶_𝑛𝑛. Then add the vertices 𝑣𝑣₁′,𝑣𝑣₂′, . . . ,𝑣𝑣_𝑛𝑛′ to the graph 𝐶𝐶_𝑛𝑛. Assign the labeling for the vertices in the following way. Define 𝑓𝑓:𝑉𝑉(𝐺𝐺)→{1,2,3, . . . ,2𝑛𝑛} as follows 𝑓𝑓(𝑣𝑣_𝑖𝑖′) =𝑛𝑛+𝑖𝑖, 1≤ 𝑖𝑖 ≤ 𝑛𝑛 and |𝑉𝑉(𝐺𝐺)| = 2𝑛𝑛, |𝐸𝐸(𝐺𝐺)| = 3𝑛𝑛. Now, we draw the splitting graph 𝑆𝑆(𝐶𝐶_𝑛𝑛) by definition 2.10. Thus |𝑒𝑒_𝑓𝑓(0)− 𝑒𝑒_𝑓𝑓(1)|≤1. Hence G is an even sum cordial graph.

Example 3.5: 𝑺𝑺(𝑪𝑪_{𝟏𝟏𝟏𝟏})

Proposition 3.6: The graph 𝐻𝐻_𝑛𝑛 is an even sum cordial graph.

Proof: Let G be a graph 𝐻𝐻_𝑛𝑛. Let 𝑥𝑥,𝑣𝑣₁,𝑣𝑣₂, . . . ,𝑣𝑣_𝑛𝑛,𝑢𝑢₁,𝑢𝑢₂, . . . ,𝑢𝑢_𝑛𝑛 be the vertices of 𝐻𝐻_𝑛𝑛. Here 𝑥𝑥 is the apex vertex of

degree n, let 𝑣𝑣₁,𝑣𝑣₂, . . . ,𝑣𝑣_𝑛𝑛 are the vertices of degree 4 and 𝑢𝑢₁,𝑢𝑢₂, . . . ,𝑢𝑢_𝑛𝑛 are the pendent vertices. Assign the labeling for the vertices in the following way. Define 𝑓𝑓:𝑉𝑉(𝐺𝐺)→{1,2,3, . . . ,2𝑛𝑛+ 1} as follows 𝑓𝑓(𝑥𝑥) = 1,𝑓𝑓(𝑣𝑣_𝑖𝑖) =𝑖𝑖+ 1,1≤ 𝑖𝑖 ≤

𝑛𝑛,𝑓𝑓(𝑢𝑢_𝑘𝑘) =𝑛𝑛+𝑘𝑘+ 1,1≤ 𝑘𝑘 ≤ 𝑛𝑛 and |𝑉𝑉(𝐺𝐺)| = 2𝑛𝑛+ 1, |𝐸𝐸(𝐺𝐺)| = 3𝑛𝑛. We draw the graph 𝐻𝐻_𝑛𝑛 from the wheel graph 𝑊𝑊_𝑛𝑛 by attaching a pendant edge to each rim vertex. Then we get

�𝑒𝑒_𝑒𝑒𝑓𝑓(0) =𝑒𝑒𝑓𝑓(1), 𝑤𝑤ℎ𝑒𝑒𝑛𝑛 𝑛𝑛 𝑖𝑖𝑖𝑖𝑒𝑒𝑣𝑣𝑒𝑒𝑛𝑛;

𝑓𝑓(0) =𝑒𝑒𝑓𝑓(1) + 1, 𝑤𝑤ℎ𝑒𝑒𝑛𝑛 𝑛𝑛 𝑖𝑖𝑖𝑖 𝑜𝑜𝑑𝑑𝑑𝑑. .

Example 3.6: (𝑯𝑯_𝟓𝟓)

Proposition 3.7: The graph 𝐹𝐹𝑙𝑙_𝑛𝑛 is an even sum cordial graph.

Proof: Let G be a graph 𝐹𝐹𝑙𝑙_𝑛𝑛. Let 𝑥𝑥,𝑣𝑣₁,𝑣𝑣₂, be the vertices of 𝐹𝐹𝑙𝑙_𝑛𝑛. Here 𝑥𝑥 is the apex vertex of degree 2𝑛𝑛, 𝑣𝑣₁,𝑣𝑣₂, . . . ,𝑣𝑣_𝑛𝑛

are the vertices of degree 4 and 𝑢𝑢₁,𝑢𝑢₂, . . . ,𝑢𝑢_𝑛𝑛 are the vertices of degree 2. Assign the labeling for the vertices in the following way. Define 𝑓𝑓:𝑉𝑉(𝐺𝐺)→{1,2,3, . . . ,2𝑛𝑛+ 1} as follows 𝑓𝑓(𝑥𝑥) = 1, 𝑓𝑓(𝑣𝑣_𝑖𝑖) =𝑖𝑖+ 1,1≤ 𝑖𝑖 ≤ 𝑛𝑛, 𝑓𝑓(𝑢𝑢_𝑘𝑘) =𝑛𝑛+

𝑘𝑘+ 1,1≤ 𝑘𝑘 ≤ 𝑛𝑛 and |𝑉𝑉(𝐺𝐺) = 2𝑛𝑛+ 1, |𝐸𝐸(𝐺𝐺)| = 4𝑛𝑛. We draw the graph 𝐹𝐹𝑙𝑙_𝑛𝑛 from the graph 𝐻𝐻_𝑛𝑛 by jining each pendant vertex of 𝐻𝐻_𝑛𝑛 to the apex vertex of 𝐻𝐻_𝑛𝑛 by an edge. Then we get 𝑒𝑒_𝑓𝑓(0) =𝑒𝑒_𝑓𝑓(1).

Thus |𝑒𝑒_𝑓𝑓(0)− 𝑒𝑒_𝑓𝑓(1)|≤1.

Hence G is an even sum cordial graph.

Example 3.7: (𝑭𝑭𝒍𝒍_𝟓𝟓)

Proposition 3.8: The graph 𝑃𝑃_𝑚𝑚(+)𝐾𝐾_𝑛𝑛 is an even sum cordial graph.

Example 3.8:

Proposition 3.9: The graph (𝐾𝐾_𝑛𝑛∪ 𝑃𝑃_𝑚𝑚) + 2𝐾𝐾₁ is an even sum cordial graph.

Example 3.9: (𝑲𝑲_𝟓𝟓∪ 𝑷𝑷_𝟔𝟔) +𝟐𝟐𝑲𝑲_𝟏𝟏

Proposition 3.10: The graph 〈𝑊𝑊_𝑛𝑛1,𝑊𝑊_𝑛𝑛2〉 is an even sum cordial graph.

Proof:Let G be a graph 〈𝑊𝑊_𝑛𝑛1,𝑊𝑊_𝑛𝑛2〉.Let 𝑢𝑢,𝑣𝑣,𝑢𝑢₁,𝑢𝑢₂, . . . ,𝑢𝑢_𝑛𝑛,𝑣𝑣₁,𝑣𝑣₂, . . . ,𝑣𝑣_𝑛𝑛 be the vertices of G. Let 𝑢𝑢₁,𝑢𝑢₂, . . . ,𝑢𝑢_𝑛𝑛 be the

rim vertices of 𝑊𝑊𝑛𝑛1 and 𝑣𝑣1,𝑣𝑣2, . . . ,𝑣𝑣𝑛𝑛 be the rim vertices of 𝑊𝑊𝑛𝑛2.

Let 𝑢𝑢 and 𝑣𝑣 be the apex vertices of 𝑊𝑊𝑛𝑛1 and 𝑊𝑊𝑛𝑛2 respectively and 𝑥𝑥 be a common vertex of 𝑊𝑊𝑛𝑛1,𝑊𝑊𝑛𝑛2. Then |𝑉𝑉(𝐺𝐺)| = 2𝑛𝑛+ 3, |𝐸𝐸(𝐺𝐺)| = 4𝑛𝑛+ 2.

Define 𝑓𝑓:𝑉𝑉(𝐺𝐺)→{1,2,3, … .2n + 3 as follows; Construction of 𝑊𝑊𝑛𝑛1. Wheel graph 𝑊𝑊𝑛𝑛1 is a graph with 𝑛𝑛+ 1 vertices formed by connecting single vertex 𝑢𝑢 to all the vertices 𝑢𝑢1,𝑢𝑢2, . . . ,𝑢𝑢𝑛𝑛 of 𝑛𝑛 cycle. Assign the label of 𝑊𝑊𝑛𝑛1 by

𝑓𝑓(𝑢𝑢𝑖𝑖) = 2𝑖𝑖, 1≤ 𝑖𝑖 ≤ 𝑛𝑛 and 𝑓𝑓(𝑢𝑢) = 1.

Construction of 𝑊𝑊𝑛𝑛2. Wheel graph 𝑊𝑊𝑛𝑛2 is a graph with 𝑛𝑛+ 1 vertices formed by connecting single vertex 𝑣𝑣 to all the vertices 𝑣𝑣1,𝑣𝑣2, . . . ,𝑣𝑣𝑛𝑛 of 𝑛𝑛 cycle. Assign the label of 𝑊𝑊𝑛𝑛2 by 𝑓𝑓(𝑣𝑣𝑖𝑖) = 2𝑖𝑖+ 1,1≤ 𝑖𝑖 ≤ 𝑛𝑛 and 𝑓𝑓(𝑣𝑣) = 2𝑛𝑛+ 2 and

𝑓𝑓(𝑥𝑥) = 2𝑛𝑛+ 3. Then connect the vertices 𝑢𝑢 and 𝑣𝑣 to the vertex 𝑥𝑥 by an edge. Then we get 𝑒𝑒𝑓𝑓(0) =𝑒𝑒𝑓𝑓(1).Thus |𝑒𝑒𝑓𝑓(0)− 𝑒𝑒𝑓𝑓(1)|≤1. Hence G is an even sum cordial graph.

Example 3.10: 〈𝑊𝑊₈1,𝑊𝑊₈2〉

Proposition 3.11: The graph𝐵𝐵_{𝑛𝑛,𝑛𝑛} is an even sum cordial graph.

Example 3.11:

Example 3.12:

4 . CONCLUSIONS

In this paper, we established the splitting graph of the family of bipartite graphs, paths and cycles are even sum cordial graphs and proved several classes of graphs such that 𝑃𝑃_𝑚𝑚(+)𝐾𝐾_𝑛𝑛,(𝐾𝐾_𝑛𝑛∪ 𝑃𝑃_𝑚𝑚) + 2𝐾𝐾₁,〈𝑊𝑊_𝑛𝑛(1),𝑊𝑊_𝑛𝑛(2)〉, 𝐵𝐵_{𝑛𝑛,𝑛𝑛},𝑆𝑆(𝐵𝐵_{𝑛𝑛,𝑛𝑛}), Helm graph 𝐻𝐻_𝑛𝑛and Flower graph 𝐹𝐹𝑙𝑙_𝑛𝑛are even sum cordial graphs.

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