Edge difference cordial labeling of path related grap ijaerdv05i0341479

Edge Difference Cordial Labeling of Path Related Graphs S. M. Vaghasiya1 , G. V. Ghodasara2 1 2

Research Scholar, R. K. University, Rajkot−360020, India.

H. & H. B. Kotak Institute of Science,Rajkot−360001, India.

1 2

sarla.spkm@gmail.com

gaurang enjoy@yahoo.co.in

————————————————————————————————————— Abstact− Let G be a (p, q) graph. Let k be an integer with 1 ≤ k ≤ q and f : E(G) → {1, 2, . . . k} be a map. For each vertex v, assign the label min|(f (ei ) − f (ej ))|. The function f is called an edge difference cordial labeling of G if f is one-to-one map and |vf (1)−vf (0)| ≤ 1 where vf (x) denotes the number of vertices labelled with x (x ∈ {1, 2, . . . , k}), ef (1) and ef (0) respectively denote the number of edges labelled with 1 and not labelled with 1. A graph with an edge difference cordial labeling is called an edge difference cordial graph. In this paper we investigate an edge difference cordial labeling for path, total graph of path, middle graph of path, splitting graph of path, square graph of path, shadow graph of path.

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Keywords: Difference cordial labeling, Edge difference cordial labeling.

Subject classification number: 05C78.

1

Introduction

We consider simple, finite, undirected graph G = (V, E). We introduced an edge difference cordial labeling of graphs. Let G be a (p, q) graph. Let k be an integer with 1 ≤ k ≤ q and f : E(G) → {1, 2, . . . k} be a map. For each vertex v, assign the label min|(f (ei )−f (ej ))|. The function f is called an edge difference cordial labeling of G if f is one-to-one map and |vf (1) − vf (0)| ≤ 1 where vf (x) denotes the number of vertices labelled with x (x ∈ {1, 2, . . . , k}), ef (1) and ef (0) respectively denote the number of edges labelled with 1 and not labelled with 1. A graph with an edge difference cordial labeling is called an edge difference cordial graph.

For the standard terminology and notations we follow Harary[1].

2

Main Results

Theorem 2.1. Path Pn admits edge difference cordial labeling except for n ≡ 0(mod4).

Proof. Let E(Pn ) = {e1 , e2 . . . , en−1 }.

We define labeling function f : E(Pn ) → {1, 2, . . . , n − 1} as follows.

f (e4i+1 ) = 4i + 1; f (e4i+2 ) = 4i + 2; f (e4i+3 ) = 4i + 4; f (e4i ) = 4i − 1;

n−2 0≤i≤ .

4

n−3 0≤i≤ .

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n−4 0≤i≤ .

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n−1 1≤i≤ .

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Here Pn satisfies the conditions for edge difference cordial labeling.

Hence Pn is an edge difference cordial graph.

Example 2.2. Edge difference cordial labeling of P7 is shown in Figure 1.

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Figure 1

Definition 2.3. The middle graph M (G) of a graph G is the graph whose vertex set is V (G) ∪ E(G) and in which two vertices are adjacent if and only if either they are adjacent edges of G or one is a vertex of G and the other is an edge incident with it.

Theorem 2.4. Middle graph of path M (Pn ) admits edge difference cordial labeling, n > 2.

Proof. Let V (Pn ) = {v1 , v2 , . . . , vn }.

0 0 are }, where the vertices v10 , v20 , . . . , vn−1 Then V (M (Pn )) = {v1 , v2 , . . . , vn , v10 , v20 , . . . , vn−1 added corresponding to edges of E(Pn ).

We define labeling function f : E(M (Pn )) → {1, 2, . . . , (3n − 4)} as follows.

Case 1: n is odd. f (vi vi0 ) = 2i; 1 ≤ i ≤ (n − 1). f (vi0 vi+1 ) = 2i − 1; 1 ≤ i ≤ (n − 1). ( 2(n + i) − 3; 1 ≤ i ≤ n−1 .

0 2 f (vi0 vi+1 ) = n−1 4(n − 1) − 2i; 2 + 1 ≤ i ≤ (n − 2).

Example 2.5. The labeling defined above is demonstrated in Figure 2.

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Figure 2 : Labeling in M (P5 )

Case 2: n is even. v1 v10 0 f (vi+1 vi+1 ) 0 f (vi vi+1 ) 0 vn vn−1

= 1. = 2i; 1 ≤ i ≤ (n − 2). = 2i + 1; 1 ≤ i ≤ (n − 2). = 2n − 2.

( 2(n + i) − 3; 1 ≤ i ≤ n−1 .

0 0 2 f (vi vi+1 ) = n−1 4(n − 1) − 2i; 2 + 1 ≤ i ≤ (n − 2).

Example 2.6. The labeling defined above is demonstrated in Figure 3.

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Figure 3 : Labeling in M (P6 ) Therefore in each case M (Pn ) satisfies the conditions for edge difference cordial labeling.

Hence M (Pn ) is an edge difference cordial graph.

Definition 2.7. The total graph of a graph G is the graph whose vertex set is V (G) ∪ E(G) and two vertices are adjacent whenever they are either adjacent or incident in G. The total graph of G is denoted by T (G).

Theorem 2.8. T (Pn ) admits edge difference cordial labeling.

Proof. Let V (Pn ) = {v1 , v2 , . . . , vn } and E(Pn ) = {e1 , e2 , . . . , en−1 }.

0 0 are }, where the vertices v10 , v20 , . . . , vn−1 Then V (T (Pn )) = {v1 , v2 , . . . , vn , v10 , v20 , . . . , vn−1 added corresponding to edges of E(Pn ).

We define labeling function f : E(T (Pn )) → {1, 2, . . . , (4n − 5)} as follows. f (ei ) = i; 1 ≤ i ≤ n − 1. (

3(n − 1) + 2i; 1 ≤ i ≤ n−3 .

0 2 n−3 f (ei ) = (5n − 6) − 2i; 2 + 1 ≤ i ≤ n − 2. (

(n − 3) + 4i; 1 ≤ i ≤ n2 .

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f (vi vi ) = (5n − 2) − 4i; n2 + 1 ≤ i ≤ n − 1. (

(n − 1) + 4i; 1 ≤ i ≤ n−1 .

0 2 n−1 f (vi+1 vi ) = 5n − 4(1 + i); 2 + 1 ≤ i ≤ n − 1.

Therefore T (Pn ) satisfies the conditions for edge difference cordial labeling.

Hence T (Pn ) is an edge difference cordial graph.

Example 2.9. Edge difference cordial labeling of T (P5 ) is shown in Figure 4.

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Figure 4

Definition 2.10. The splitting graph of a graph G is obtained by adding to each vertex v, a new vertex v 0 such that v 0 is adjacent to every vertex that is adjacent to v in G. The splitting graph of G is denoted by S(G).

Theorem 2.11. S(Pn ) admits an edge difference cordial labeling for n > 2.

Proof. Let V (Pn ) = {v1 , v2 , . . . , vn }.

0 0 Then V (S(Pn )) = {v1 , v2 , . . . , vn , v10 , v20 , . . . , vn−1 }, where the vertices v10 , v20 , . . . , vn−1 are added corresponding to V (Pn ).

We define labeling function f : E(M (Pn )) → {1, 2, . . . , (3n − 3)} as follows.

Case 1: n is odd. f (vi vi+1 ) = i; 1 ≤ i ≤ (n − 1). v1 v20 = n + 1. ( (n − 4) + 4i; 2 ≤ i ≤ n−1 .

0 2 f (vi vi+1 ) = n−1 (5n − 1) − 4i; 2 + 1 ≤ i ≤ (n − 1). ( . (n − 2) + 4i; 1 ≤ i ≤ n−1 2 f (vi0 vi+1 ) = n−1 (5n − 3) − 4i; 2 + 1 ≤ i ≤ (n − 2).

0 vn−1 vn = n.

Example 2.12. The labeling defined above is demonstrated in Figure 5.

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Figure 5 Case 2: n is even. f (vi vi+1 ) = i; 1 ≤ i ≤ (n − 1). ( (n − 2) + 4i; 1 ≤ i ≤ n−2 .

0 2 f (vi vi+1 ) = n−2 (5n − 3) − 4i; 2 + 1 ≤ i ≤ (n − 1). ( n + 4i; 1 ≤ i ≤ n−2 .

2 f (vi0 vi+1 ) = 5(n − 1) − 4i; n−2 + 1 ≤ i ≤ (n − 2).

2 0 vn−1 vn = n.

Example 2.13. The labeling defined above is demonstrated in Figure 6.

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Figure 6 Therefore in each case S(Pn ) satisfies the conditions for edge difference cordial labeling.

Hence S(Pn ) is an edge difference cordial graph.

Definition 2.14. The square of graph G denoted by G2 , is defined as the graph with V (G) = V (G2 ), where two vertices are adjacent in G2 if they are at a distance 1 or 2 apart in G. Theorem 2.15. Pn 2 admits an edge difference cordial labeling for n > 3.

Proof. Let V (Pn 2 ) = {v1 , v2 , . . . , vn }.

We define labeling function f : E(Pn 2 ) → {1, 2, . . . , (2n − 3)} as follows.

Case 1: n is odd. n−3 .

2 n−3 4i − 1; 1 ≤ i ≤ .

2 (3n − 5) − i; (n − 2) ≤ i ≤ (n − 1). n−1 4i − 3; 1 ≤ i ≤ .

2 n−3 4i − 2; 1 ≤ i ≤ .

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f (v2i−1 v2i ) = 4i; 1 ≤ i ≤ f (v2i v2i+1 ) = f (vi vi+1 ) = f (v2i−1 v2i+1 ) = f (v2i v2i+2 ) =

Here Pn 2 satisfies the conditions for edge difference cordial labeling.

Example 2.16. The labeling defined above is demonstrated in Figure 7.

Case 2: n is even. f (vi vi+1 ) = i + 2; 1 ≤ i ≤ 2. n−4 f (v2i+1 v2i+2 ) = 4i + 4; 1 ≤ i ≤ .

2 n−4 f (v2i+2 v2i+3 ) = 4i + 3; 1 ≤ i ≤ .

2 f (vn−1 vn ) = 2n − 3. n f (v2i−1 v2i+1 ) = 4i − 3; 1 ≤ i ≤ − 1.

2 n f (v2i v2i+2 ) = 4i − 2; 1 ≤ i ≤ − 1.

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Figure 7

Here Pn 2 satisfies the conditions for edge difference cordial labeling.

Example 2.17. The labeling defined above is demonstrated in Figure 8.

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Figure 8 Therefore in each case Pn 2 satisfies the conditions for edge difference cordial labeling.

Hence Pn 2 is an edge difference cordial graph.

Definition 2.18. The shadow graph of a connected graph G denoted as D2 (G), is constructed by taking two copies of G say G0 and G00 and joining each vertex u0 in G0 to the neighbors of the corresponding vertex u00 in G00 .

Theorem 2.19. D2 (Pn ) admits an edge difference cordial labeling.

Proof. Let V (D2 (Pn )) = {v1 , v2 , . . . , v2n }.

We define labeling function f : E(D2 (Pn )) → {1, 2, . . . , 4(n − 1)} as follows.

Case 1: n is odd. f (vi vi+1 ) = i; 1 ≤ i ≤ (n − 1).

( . (n − 3) + 4i; 1 ≤ i ≤ n−1 0 2 f (vi vi+1 ) = n−1 (5n − 2) − 4i; 2 + 1 ≤ i ≤ (n − 1). ( (n − 1) + 4i; 1 ≤ i ≤ n−1 .

2 f (vi0 vi+1 ) = n−1 (5n − 4) − 4i; 2 + 1 ≤ i ≤ (n − 1). ( 3(n − 1) + 2i; 1 ≤ i ≤ n−1 .

0 0 2 f (vi vi+1 ) = + 1 ≤ i ≤ (n − 1).

(5n − 4) − 2i; n−1 2 Example 2.20. The labeling defined above is demonstrated in Figure 9.

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Figure 9 Case 2: n is even. f (vi vi+1 ) = i; 1 ≤ i ≤ (n − 1). ( (n − 3) + 4i; 1 ≤ i ≤ n2 .

0 f (vi vi+1 ) = (5n − 2) − 4i; n2 + 1 ≤ i ≤ (n − 1). ( . (n − 1) + 4i; 1 ≤ i ≤ n−2 2 f (vi0 vi+1 ) = n−2 (5n − 4) − 4i; 2 + 1 ≤ i ≤ (n − 1). ( (3n − 4) + 2i; 1 ≤ i ≤ n2 .

0 0 f (vi vi+1 ) = (5n − 3) − 2i; n2 + 1 ≤ i ≤ (n − 1).

Example 2.21. The labeling defined above is demonstrated in Figure 10.

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Figure 10 In each case D2 (P6 ) satisfies the conditions for edge difference cordial labeling.

Hence D2 (P6 ) is an edge difference cordial graph.

References [1] F. Harary, Graph theory, Addision-wesley, Reading, MA (1969).

[2] I. Cahit, On cordial and 3-equitable labelings of graphs, Util. Math., 37(1990), 189-198.

[3] J. A. Gallian, A dynemic survey of graph labeling, The Electronics Journal of Combinatorics, 16(2013), ]DS6 1 − 308.

[4] J. Gross and J. Yellen, Graph theory and its applications, CRC Press, (1999).

[5] R. Ponraj, S. Sathish Narayanan and R. Kala, Difference Cordial Labeling of Graphs, Global Journal of Mathematical Sciences: Theory and Practical, 5 (2013) 185-196.

[6] R. Ponraj, M. Maria Adaickalam and R. Kala, k-difference cordial labeling of graphs, International Journal of Mathematical Combinatorics, 2 (2016), 121-131.

[7] R. Ponraj and M. Maria Adaickalam, 3-difference cordial labeling of some cycle related graphs, Journal of Algorithms and Computation, 47 (2016), 1-10.

[8] S. M. Vaghasiya and G. V. Ghodasara, Difference Cordial of Operational Graph Related to Cycle, International Journal of Advanced Engineering Research and Science, 3 (2016), 236-239.

[9] S. M. Vaghasiya and G. V. Ghodasara, Edge Difference Cordial Labeling of Graphs, Accepted for publication in International Journal of Mathematics And Its Applications.