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## Print version ISSN 0716-0917

### Proyecciones (Antofagasta) vol.36 no.4 Antofagasta Dec. 2017

#### http://dx.doi.org/10.4067/S0716-09172017000400601

Articles

Some results on SD-Prime cordial labeling

1St. Xavier's College (Autonomous), Department of Mathematics, Palayamkottai, Tamilnadu, India, e-mail : lourdusamy15@gmail.com

2St. Xavier's College (Autonomous), Department of Mathematics, Palayamkottai, Tamilnadu, India, e-mail : patrick881990@gmail.com

Abstract:

Given a bijection ʄ: V(G) → {1,2, …,|V(G)|}, we associate 2 integers S = ʄ(u)+ʄ(v) and D = |ʄ(u)-ʄ(v)| with every edge uv in E(G). The labeling ʄ induces an edge labeling ʄ' : E(G) → {0,1} such that for any edge uv in E(G), ʄ '(uv)=1 if gcd(S,D)=1, and ʄ ' (uv)=0 otherwise. Let ' (i) be the number of edges labeled with i ∈ {0,1}. We say ʄ is SD-prime cordial labeling if | eʄ ' (0)- e ʄ' (1)| ≤ 1. Moreover G is SD-prime cordial if it admits SD-prime cordial labeling. In this paper, we investigate the SD-prime cordial labeling of some derived graphs.

Keywords : SD-prime labeling; SD-prime cordial labeling; star.

1. Introduction

Let G=(V(G),E(G)) be a simple, finite and undirected graph of order |V(G)|=p and size |E(G)|=q. All notations not defined in this paper can be found in (2). A labeling of a graph is a map that carries the graph elements to the set of numbers, usually to the set of non-negative or positive integers. If the domain is the set of vertices the labeling is called vertex labeling. If the domain is the set of edges, then we called about edge labeling. If the labels are assigned to both vertices and edges then the labeling is called total labeling. For all detailed survey of graph labeling we to refer Gallian (1). In (4) (5), G.C. Lau and W.C. Shiu have introduced the concepts SD-prime labeling. In (3), G. C. Lau et.al. have introduced SD-prime cordial labeling and they proved behaviour of several graphs like path, complete bipartite graph, star, double star, wheel, fan, double fan and ladder are SD-prime cordial labeling. In this paper, we investigate the SD-prime cordial labeling behavior of S'(K1,n), D2(K1,n), S(K1,n), DS(K1,n), S'(Bn,n), D2(Bn,n), DS(Bn,n), S(Bn,n), K1,3 * K1,, CHn, Fln, , T(Pn), T(Cn), the graph obtained by duplication of each vertex of path and cycle by an edge, Qn, A(Tn), TLn, Pn ʘ K1, Cn ʘ K1 and Jn.

In (4), Lau and Shiu introduced a variant of prime graph labeling which is defined as follows.

Given a bijection ʄ: V(G) → {1,2,...,|V(G)|}, we associate 2 integers S = ʄ(u)+ ʄ(v) and D =|ʄ(u)-ʄ(v)| with every edge uv in E.

Definition 1.1. (4) A bijection ʄ: V(G) → {1,2,...,|V(G)|}, induces an edge labeling ʄ ': E(G) → {0,1} such that for any edge uv in G, ʄ '(uv)=1 if gcd(S,D)=1, and ʄ '(uv)=0 otherwise. We say ʄ is SD-prime labeling if ʄ '(uv)=1 for all uvE(G). Moreover, G is SD-prime if it admits SD-prime labeling.

Definition 1.2.(3) A bijection ʄ: V(G) → {1,2,...,|V(G)|}, induces an edge labeling ʄ ': E(G) → {0,1} such that for any edge uv in G, ʄ '(uv)=1 if gcd(S,D)=1, and ʄ '(uv)=0 otherwise. The labeling ʄ is called SD-prime cordial labeling if |e ʄ '(0)- e ʄ '(1)| ≤ 1. We say that G is SD-prime cordial if it admits SD-prime cordial labeling.

Definition 1.3. For every vertex vV(G), the open neighbourhood set N(v) is the set of all vertices adjacent to v in G.

Definition 1.4. For a graph G the splitting graph S'(G) of a graph G is obtained by adding a new vertex v' corresponding to each vertex v of G such that N(v)=N(v').

Definition 1.5. The shadow graph D2(G) of a connected graph G is obtained by taking two copies of G, say G' and G''. Join each vertex u' in G' to the neighbours of corresponding vertex u'' in G''.

Definition 1.6. Duplication of a vertex vk by a new edge e = vk' vk'' in a graph G produces a new graph G' such that N( vk' ) ⋂ N( vk'' ) = vk .

Definition 1.7. Let G be the a graph with V = S1 S2 S3 ⋃ ... St T where each Si is a set of vertices having at least two vertices of the same degree and T=V(G) \ Si . The degree splitting graph of G denoted by DS(G) is obtained from G by adding vertices w1,w2,...,wt and joining to each vertex of Si for 1 ≤ it.

Definition 1.8. For a simple connected graph G the square of graph G is denoted by G2 and defined as the graph with the same vertex set as of G and two vertices are adjacent in G2 if they are at a distance 1 or 2 apart in G.

Definition 1.9. The subdivision graph S(G) is obtained from G by subdividing each edge of G with a vertex.

Definition 1.10. The bistar Bn,n is the graph obtained by attaching the apex vertices of two copies of K{1,n} by an edge.

Definition 1.11. K1,3 * K1,n is the graph obtained from K 1,3 by attaching root of a star K 1,n at each pendant vertex of K 1,3.

Definition 1.12. The triangular ladder is a graph obtained from Ln by adding the edges uivi+1 , 1 ≤ in-1, where ui and vi , 1 ≤ in, are the vertices of Ln such that u1, u2 ,..., un and v1, v2,...,vn are two paths of order n in the graph Ln .

Definition 1.13. The corona G1 ʘ G2 of two graphs G1 ( p1 , q1 ) and G2 ( p2 , q2 ) is defined as the graph obtained by taking one copy of G1 and p1 copies of G2 and joining the ith vertex of G1 with an edge to every vertex in the ith copy of G2 .

Definition 1.14. The total graph T(G) 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.

2. Main results

Theorem 2.1. If G is SD-prime cordial of size q, then G - e is also SD-prime cordial

(i) for all eE(G) when q is even.

(ii) for some eE(G) when q is odd.

Proof. Case (i): when q is even.

Let G be the SD-prime cordial graph of size q, where q is an even number. It follows that e ʄ '(0)= e ʄ '(1)= . Let e be any edge in G which is labeled either 0 or 1. Then in G - e, we have either e ʄ '(0)= e ʄ '(1)+1 or e ʄ '(1)= e ʄ '(0)+1 and hence | e ʄ '(0) - e ʄ '(1)| ≤ 1. Thus G - e is SD-prime cordial for all eE(G).

Case (ii): when q is odd.

Let G be the SD-prime cordial graph of size q, where q is an odd number. It follows that either e ʄ '(0)= e ʄ '(1)+1 or e ʄ '(1)= e ʄ '(0)+1. If e ʄ '(0)= e ʄ '(1)+1 then remove an edge e which is labeled as 0 and if e ʄ '(1)= e ʄ '(0)+1 then remove an edge e which is labeled as 1 from G. It follows that e ʄ '(0)= e ʄ '(1). Thus, G - e is SD-prime cordial for some eE(G).

Corollary 2.2. The graph G + e is SD-prime cordial if G is SD-prime cordial having even size.

Theorem 2.3. The graph S'( K1,n ) is SD-prime cordial.

Proof. Let v1, v2, ..., vn be the pendant vertices and v be the apex vertex of K1,n and u, u1, u2,..., un are added vertices corresponding to v,v 1, v2,...,vn to obtain S'( K1,n ). Therefore, S'( K1,n ) is of order 2n+2 and size 3n. Define ʄ: V(S'( K1,n ))→ {1,2,...,2n+2} as follows:

Theorem 2.4. The graph D2 (K1,n) is SD-prime cordial.

Proof.Let u, u1, u2,..., un and v, v 1, v2,...,vn be the vertices of two copies of K1,n . Let V(D2(K1,n)) = {u, v} ⋃ { ui, v i: 1 ≤ in} and E(D2(K1,n)) = uui , vvi , uvi , vui : 1 ≤ in}. Therefore, D2 ( K1,n ) is of order 2n+2 and size 4n. Define ʄ: V( D2 ( K1,n ))→ {1,2,...,2n+2} as follows:

In view of the above labeling pattern we get, eʄ ' (0) = eʄ ' (1)= 2n.

Thus, |e ʄ '(0)- e ʄ '(1)| ≤ 1. Hence, D2 ( K1,n ) is SD-prime cordial.

Theorem 2.5. The graph S( K1,n ) is SD-prime cordial.

Proof. Let v, v 1, v2,...,vn be the vertices of K 1,n. Let V(S( K1,n )) = {v} ⋃ { vi,ui : 1 ≤ in} and E(S( K1,n )) = { vui, viui : 1 ≤ in}. Therefore, S( K1,n ) is of order 2n+1 and size 2n. Define ʄ: V( S2 ( K1,n ))→ {1,2,...,2n+1} as follows:

In view of the above labeling pattern we get, e ʄ '(0) = e ʄ '(1) = n.

Thus, | e ʄ '(0) - e ʄ '(1)| ≤ 1. Hence S( K1,n ) is SD-prime cordial.

Theorem 2.6. The graph S'( Bn,n ) is SD-prime cordial.

Proof. Let u1, u2,..., un, v1, v2 ,...,vn be the pendant vertices and u,v be the apex vertices of Bn,n. Let u', v', u'i, v'i be added vertices corresponding to u, v ,ui, vi to obtain S'( Bn,n ). Therefore, S'( Bn,n ) is of order 4n+4 and size 6n+3. Define ʄ: V(S'( Bn,n ))→ {1,2,...,4n+4} as follows:

Thus, | e ʄ '(0) - e ʄ '(1)| ≤ 1. Hence, S'( Bn,n ) is SD-prime cordial.

Theorem 2.7. The graph D2(Bn,n) is SD-prime cordial.

Proof. Let u1, u2,..., un, v1, v2 ,...,vn be the pendant vertices and u,v be the apex vertices of Bn,n. Let

V( D2 ( Bn,n )) 0 = {u, v, u',v'} ⋃ { ui, vi, u'i, v'I : 1 ≤ in} and E(D2(Bn,n)) = {uv, u'v, uv',u'v' } ⋃ { uui, uu'i, u' u'i, u' ui, vvi, vv'i, v' vi, v' v'i : 1 ≤ in }.

Therefore, D2 ( Bn,n ) is of order 4n+4 and size 8n+4. Define ʄ: V( D2 ( Bn,n )) → {1,2, …, 4n+4} as follows:

Let p be the highest prime number <4n+4.

For the vertices u1, u2,..., un, v'1, v'2, ... ,v'n we assign distinct odd numbers except 1 and p. In view of the above labeling pattern we get, e ʄ '(0)= e ʄ '(1) = 4n+2.

Thus, |e ʄ '(0)- e ʄ '(1)| ≤ 1. Hence, D2 ( Bn,n ) is SD-prime cordial.

Theorem 2.8. The graph DS(B n,n) is SD-prime cordial.

Proof. Let u1, u2, ... , un, ,v1 ,v2, ... ,vn be the pendant vertices and u,v be the apex vertices of Bn,n . Let V(DS( Bn,n )) = { u, v, w1, w2 } ⋃ {ui ,vi: 1 ≤ in } and E(DS( Bn,n ))= {uv, uw2, vw2} ⋃ { uui, vvi, ui w1,viw1 : 1 ≤ in }. Therefore, DS(Bn,n) is of order 2n+4 and size 4n+3. Define ʄ: V(DS( Bn,n )) → {1,2, …, 2n+4} as follows:

In view of the above labeling pattern we get, e ʄ '(0)= 2n+1. and e ʄ '(1)= 2n+2.

Thus, |e ʄ '(0) - e ʄ '(1)| ≤ 1. Hence, DS(Bn,n) is SD-prime cordial.

Theorem 2.9. The graph S(Bn,n) is SD-prime cordial.

Proof. Let u1, u2, ... , un, ,v1 ,v2, ... ,vn be the pendant vertices and u,v be the apex vertices of Bn,n .. Let v'i and u'i be the newly added vertices between v and vi and u and ui respectively. Also, w be the newly added vertex between u and v. Therefore, S(Bn,n) is of order 4n+3 and size 4n+2. Define ʄ: V(S( Bn,n ))→ {1,2,...,4n+3} as follows:

In view of the above labeling pattern we get, e ʄ '(0)= e ʄ '(1) = 2n+1.

Thus, |e ʄ '(0) - e ʄ '(1)| ≤ 1. Hence, S(Bn,n) is SD-prime cordial.

Theorem 2.10. The graph K1,3 * K1,n is SD-prime cordial.

Proof. Let V( K1,3 * K1,n )= {x, u, v, w} ⋃ {ui, vi, wi: 1 ≤ in } and E(K 1,3 * K1,n )= {xu, xv, xw} ⋃ { uui, vvi, wwi: 1 ≤ i ≤ n }. Therefore, K1,3 * K1,n is of order 3n+4 and size 3n+3. Define ʄ: V( K1,3 * K1,n ) → {1,2, … ,3n+4} as follows:

In view of the above labeling pattern we get,

Thus, |e ʄ '(0) - e ʄ '(1)| ≤ 1. Hence, K1,3 * K1,n is SD-prime cordial.

Theorem 2.11. The triangular ladder TL n is SD-prime cordial.

Proof. Let v1 ,v2, ... ,vn , u1, u2, ... , un be the vertices of Ln . Let V(TLn) = { ui ,vi: 1 ≤ in } and E(TLn) = { uiui+1, vivi+1, uivi+1 : 1 ≤ i ≤ n-1 } ⋃ { uivi }:1 ≤ in}. Therefore, TLn is of order 2n and size 4n-3. Define ʄ: V( TLn )→{1,2, … ,2n} as follows:

In view of the above labeling pattern we get, e ʄ '(0)= 2n - 2 and e ʄ '(1)= 2n - 1.

Thus, | e ʄ '(0) - e ʄ '(1)| ≤ 1. Hence, TLn is SD-prime cordial.

Theorem 2.12. The closed helm graph CHn is SD-prime cordial.

Proof. Let v be an apex vertex and v1, v2, …, vn are rim vertices of Wn . Let u1 , u2 , … , un be the pendant vertices which are joined to each rim vertices of Wn to obtained Hn . So V( CHn )= {v} ⋃ { vi, ui : 1 ≤ in } and E(CHn) = { vvi , uivi : 1 ≤ in} ⋃ { vivi+1, uiui+1 : 1 ≤ in-1 } ⋃ { v1vn, u1un }. Therefore, CHn is of order 2n+1 and size 4n. Define ʄ: V(CHn) → {1,2, … ,2n+1} as follows:

In view of the above labeling pattern we get, e ʄ '(0)= e ʄ '(1)= 2n.

Thus, | e ʄ '(0) - e ʄ '(1)| ≤ 1. Hence, CHn is SD-prime cordial.

Theorem 2.13. The flower Fln is SD-prime cordial.

Proof. Let V(Fln) = {v} ⋃ { vi, ui : 1 ≤ in } and E(Fln) = { vvi, viui, vui : 1 ≤ in } ⋃ { vnv1 } ⋃ { vivi+1 : 1 ≤ in-1}. Therefore, Fln is of order 2n+1 and size 4n. Define ʄ: V( Fln ) → {1,2, … ,2n+1} as follows. Assign the labels to vertices v, vi, ui as in theorem 2.14. We observe that, e ʄ '(0) = e ʄ '(1) = 2n.

Thus, | e ʄ '(0) - e ʄ '(1)| ≤ 1.. Hence, Fln is SD-prime cordial.

Theorem 2.14. The graph is SD-prime cordial.

Proof. Let v1, v2, ..., vn be the vertices of the path Pn . Let V( )= { v1,v2,...,vn } and E( )= { vivi+1 : 1 ≤ in-1 } ⋃ { vivi+2 : 1 ≤ in-2}. Therefore, is of order n and size 2n-3. We define ʄ: V( → {1,2,...,n} by ʄ ( vi )= i for 1 ≤ in. We observe that, e ʄ '(0) = n-2 and e ʄ '(1)= n - 1.

Thus, | e ʄ '(0) - e ʄ '(1)| ≤ 1. Hence, is SD-prime cordial.

Remark 2.15. Note that T(Pn) = . Therefore, T(Pn) is also SD-prime cordial.

Theorem 2.16. The graph obtained by duplication of each vertex by an edge in Pn is SD-prime cordial.

Proof. Let v1, v2, ..., vn be the vertices of the path Pn and G be the graph obtained by duplication of each vertex vi of the path Pn by an edge vi'vi'' for 1 ≤ in at a time. Let V(G)= { vi, vi', vi'' : 1 ≤ in} and E(G)= { vivi',vivi'',vi'vi'' : 1 ≤ in } ⋃ { vivi+1 : 1 ≤ in-1}. Therefore, G is of order 3n and size 4n-1. Define ʄ: V(G) → {1,2,...,3n} as follows:

In view of the above labeling pattern we get, e ʄ '(1)= 2n and e ʄ '(0)= 2n-1.

Thus, | e ʄ '(0) - e ʄ '(1)| ≤ 1. Hence, G is SD-prime cordial.

Theorem 2.17 The graph T(Cn) is SD-prime cordial.

Proof. Let v1, v2, … ,vn be the vertices of the cycle Cn . Let V(T(Cn)) = { vi, ui : 1 ≤ in} and E(T(Cn))= { vivi+1, uiui+1 : 1 ≤ in-1} ⋃ { viui : 1 ≤ in } ⋃ { viui-1 : 2 ≤ in } ⋃ { vnv1,unu1,v1un }. Therefore, T(Cn) is of order 2n and size 4n. Define ʄ: V(T(Cn)) → {1,2, …,2n} as follows:

In view of the above labeling pattern we get, e ʄ '(0)= e ʄ '(1)= 2n.

Thus, | e ʄ '(0) - e ʄ '(1)| ≤ 1. Hence, T(Cn) is SD-prime cordial.

Remark 2.18. Note that T(Cn) = . It is natural to determine the SD-prime cordiality of , n ≥ 2.

Theorem 2.19. The graph obtained by duplication of each vertex by an edge in Cn is SD-prime cordial.

Proof. Let v1, v2,...,vn be the vertices of the cycle Cn and G be the graph obtained by duplication of each vertex vi of the cycle Cn by an edge v'iv''i for 1in. Then V(G)= {vi, vi',vi'': 1 ≤ in} and

E(G)= E(Cn) viv'i, viv''i, v'iv''i : 1 ≤ in }. Therefore, G is of order 3n and size 4n. Define ʄ: V(G)→ {1,2,...,3n} as follows:

In view of the above labeling pattern we get, e ʄ '(0)= e ʄ '(1)= 2n.

Thus, | e ʄ '(0) - e ʄ '(1)| ≤ 1. Hence, G is SD-prime cordial.

Theorem 2.20. The quadrilateral snake Qn is SD-prime cordial.

Proof. Let v1,v2, … ,vn be the vertices of path Pn . Let V(Qn)= V(Pn) ⋃ {ui, wi: 1 ≤ in-1} and E(Qn)=E(Pn) ⋃ { viui,uiwi,vi+1wi : 1 ≤ in-1}. Therefore, Qn is of order 3n-2 and size 4n-4. Define ʄ: V(Qn) → {1,2, … ,3n-2} as follows:

In view of the above labeling pattern we get, e ʄ '(0)= e ʄ '(1)= 2n-2.

Thus, | e ʄ '(0) - e ʄ '(1)| ≤ 1. Hence, Qn is SD-prime cordial.

Theorem 2.21. An alternate triangular snake A(Tn) is SD-prime cordial.

Proof. Let v1, v2, … , vn be the vertices of path Pn . The graph A(Tn) is obtained by joining the vertices vivi+1 (alternately) to a new vertex ui, 1 ≤ in-1 for even n and 1 ≤ in-2 for odd n. Therefore, V(A(Tn))=V(Pn) ⋃ {ui}:1 ≤ i ≤ ⌊ ⌋}{2} and E(A(Tn)) = E(Pn) ⋃ { v2i-1ui : 1 ≤ i ≤ ⌊ ⌋ ; uiv2i : 1 ≤ i ≤ ⌊ ⌋}. Also,

Case 1: n is odd.

We observe that, e ʄ '(0)= e ʄ '(1)= n-1.

Case 2: n is even.

We observe that, e ʄ '(0)= n -1 and e ʄ '(1)= n.

Thus, in both cases | e ʄ '(0) - e ʄ '(1)| ≤ 1. Hence, A(Tn) is SD-prime cordial.

Theorem 2.22. The comb Pn ʘ K1 is SD-prime cordial.

Proof. Let v1, v2, ... , vn be the vertices of path Pn . Let V(Pn ʘ K1) = { vi, ui : 1 ≤ in} and E(Pn ʘ K1) = { vivi+1 : 1 ≤ in-1} ⋃{ viui : 1 ≤ in}. Therefore, Pn ʘ K1 is of order 2n and size 2n-1. Define ʄ: V( Pn ʘ K1 ) → {1,2,...,2n} as follows:

In view of the above labeling pattern we get, e ʄ '(0)= n - 1 and e ʄ '(1)= n.

Thus, | e ʄ '(0) - e ʄ '(1)| ≤ 1. Hence, Pn ʘ K1 is SD-prime cordial.

Corollary 2.23. The crown Cn ʘ K1 is SD-prime cordial.

Theorem 2.24. The jewel Jn is SD-prime cordial.

Proof. Let V(Jn)= {u,v,x,y} ⋃ { ui : 1 ≤ in} and E(Jn) = {ux,uy,xy,vx,vy} ⋃ { uui,vui: 1 ≤ in }. Therefore, Jn is of order n+4 and size 2n+5. Define ʄ: V(Jn) → {1,2, … ,n+4} as follows:

In view of the above labeling pattern we get, e ʄ '(0)= n + 2 and e ʄ '(1)= n + 3.

Thus, | e ʄ '(0) - e ʄ '(1)| ≤ 1. Hence, Jn is SD-prime cordial.

References

[1] J. A. Gallian, A Dyamic Survey of Graph Labeling, The Electronic J. Combin., 17, (2014) \# DS6. [ Links ]

[2] F. Harary, Graph Theory, Addison-Wesley, Reading, Mass, (1972). [ Links ]

[3] G. C. Lau, H. H. Chu, N. Suhadak, F. Y. Foo and H. K. Ng, On SD-Prime Cordial Graphs, International Journal of Pure and Applied Mathematics, 106 (4), pp. 1017-1028, (2016). [ Links ]

[4] G. C. Lau and W. C. Shiu, On SD-Prime Labeling of Graphs, Utilitas Math., accepted. [ Links ]

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Received: February 2017; Accepted: March 2017

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