4-Prime cordiality of some cycle related graphs

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1 Available at Appl. Appl. Math. ISSN: Vol. 1, Issue 1 (Jue 017), pp Applicatios ad Applied Mathematics: A Iteratioal Joural (AAM) 4-Prime cordiality of some cycle related graphs 1 R. Poraj, Rajpal Sigh, & 1 S. Sathish Narayaa 1 Departmet of Mathematics Sri Paramakalyai College Alwarkurichi-6741, Idia porajmaths@gmail.com; sathishrvss@gmail.com & Research Scholar Departmet of Mathematics Maomaiam Sudaraar Uiversity Abishekapatti, Tiruelveli , Tamiladu, Idia rajpalsigh@outlook.com Received: Jue 18, 01; Accepted: October 0, 016 Abstract Recetly three prime cordial labelig behavior of path, cycle, complete graph, wheel, comb, subdiviso of a star, bistar, double comb, coroa of tree with a vertex, crow, olive tree ad other stadard graphs were studied. Also four prime cordial labelig behavior of complete graph, book, flower were studied. I this paper, we ivestigate the four prime cordial labelig behavior of coroa of wheel, gear, double coe, helm, closed helm, butterfly graph, ad friedship graph. Keywords: Wheel; Gear; Double Coe; Helm; Butterfly Graph MSC 010 No.: Primary 05C78, Secodary 05C38 1. Itroductio Throughout this paper we have cosidered oly simple ad udirected graph. Terms ad defiitios ot defied here are used i the sese of Harary (001) ad Gallia (015). 30

2 AAM: Iter. J., Vol. 1, Issue 1 (Jue 017) 31 Let G = (V,E) be a (p,q) graph. The cardiality of V is called the order of G ad the cardiality of E is called the size of G. Rosa (1967) itroduced graceful labellig of graphs which was the foudatio of the graph labellig. Cosequetly Graham (1980) itroduced harmoious labellig. Cahit (1987) iitiated the cocept of cordial labellig of graphs. Ebrahim Salehi (010) defied the otio of product cordial set. Salehi ad Mukhi (01) said a graph G of size q is fully product-cordial if its product cordial set is {q k : 0 k q }. Because the product-cordial set is the multiplicative versio of the friedly idex set, Kwog, Lee, ad Ng (010) called it the product-cordial idex set of G ad determied the exact values of the product-cordial idex set of Cm ad C m P. Kwog et al. (01), determied the friedly idex sets ad product-cordial idex sets of -regular graphs ad the graphs obtaied by idetifyig the ceters of ay umber of wheels. Shiu (013) determied the product-cordial idex sets of grids C m P. Sudaram et al. (005) itroduced the cocept of prime cordial labellig of graphs. Seoud ad Salim (010) gave a upper boud for the umber of edges of a graph with a prime cordial labelig as a fuctio of the umber of vertices. For bipartite graphs they gave a stroger boud. O aalogous of this, the otio of k-prime cordial labellig has bee itroduced by Poraj et al. (016). Poraj et al. (016) studied the k-prime cordial labellig behaviour of paths, cycles, bistars of eve order. Also they studied about the 3-prime cordiality of paths, cycles, coroa of tree with a vertex, comb, crow, olive tree ad some more graphs. I this paper, we ivestigate the 4-prime cordial labellig behaviour of wheel, gear, double coe, helm, butterfly graph ad closed helm.. Prelimiaries Let f : V (G) {1,,...,k} be a fuctio. For each edge uv, assig the label gcd( f (u), f (v)). f is called k-prime cordial labellig of G if v f (i) v f ( j) 1, i, j {1,,...,k} ad e f (0) e f (1) 1 where v f (x) deotes the umber of vertices labelled with x, e f (1) ad e f (0) respectively the umber of edges labeled with 1 ad ot labelled with 1. A graph with a k-prime cordial labellig is called a k-prime cordial graph. Let G 1 ad G be two graphs with vertex sets V 1 ad V ad edge sets E 1 ad E respectively. The the joi G 1 +G is the graph whose vertex set is V 1 V ad edge set is give by E 1 E {uv : u V 1 ad v V }. The graph C + K 1 is called the wheel. I a wheel, the vertex of degree is called the cetral vertex ad the vertices o the cycle C are called rim vertices. The graph C + K is called a double coe. A gear graph is obtaied from the wheel W by addig a vertex betwee every pair of adjacet

3 3 R. Poraj et al. vertices of the -cycle. The helm H is the graph obtaied from a wheel by attachig a pedet edge at each vertex of the -cycle. A closed helm CH as the graph obtaied from a helm by joiig each pedet vertex to form a cycle. Two eve cycles of the same order, say C, sharig a commo vertex with m pedet edges attached at the commo vertex is called a butterfly graph By m,. The oe-poit uio of t copies of the cycle C 3 is called a friedship graph C (t) 3. Remark. A -prime cordial labelig is a product cordial labelig. [Sudaram et al. (004)] I the ext sectio we ivestigate the 4-prime cordiality of some graphs. 3. Mai results First we ivestigate the 4-prime cordiality of the wheel W ad a double coe C + K. Theorem 1. The wheel W is a 4-prime cordial if ad oly if 3,4,7. Let C : u 1 u...u u 1 be the cycle ad let V (K 1 ) = {u}. Obviously W has + 1 vertices ad edges. First we cosider the case whe = 3. Clearly it is isomorphic to K 4. Sice there are four vertices, each should be labelled with differet labels from the set {1,,3,4}. If it is so, the e f (0) = 1 ad e f (1) = 5. This gives e f (1) e f (0) > 1, so there does ot exists a 4-prime cordial labellig f of W 3. For W 4 let g be a 4-prime cordial labellig. The v g (1) =, v g () = v g (3) = v g (4) = 1 (or) v g () =, v g (1) = v g (3) = v g (4) = 1 (or) v g (3) =, v g (1) = v g () = v g (4) = 1 (or) v g (4) =, v g (1) = v g () = v g (3) = 1. Clearly it is ot too hard to show that eg (1) e g (0) 3. So W 4 does ot admit ay 4-prime cordial labellig. For completig the proof of oexistece part, it remais to show that W 7 does ot permit ay 4-prime cordial labelig. Here each label must be used exactly twice. The the umber of edges ot labelled 1 is at most 6. Thus there does ot exist a 4-prime cordial labellig for W 7. This completes the oexistece part. Now we cosider the existece part. We divide the proof ito four cases. Case 1. 0 (mod 4) ad 4. I this case we ca take the value of be of the form 4t, t > 1. We costruct a vertex labellig

4 AAM: Iter. J., Vol. 1, Issue 1 (Jue 017) 33 h for this case as follows: Assig the label to the vertices u 1, u,..., u t ad u. The vertices u t+1, u t+,..., u t should be labelled with 4. The the ext two cosecutive vertices u t+1, u t+ are labelled with 3. Use the label 1 to the vertices u t+3 ad u t+4. Now, from u t+5 to u 4t, we put the labels 3, 1 alterately as h(u t+5 ) = 3, h(u t+6 ) = 1, ad so o. It is easy to check that h(u 4t ) = 1. From the above labellig h, we have the vertex ad edge coditios respectively are v h () = t + 1, v h (1) = v h (3) = v h (4) = t ad e h (0) = e h (1) = 4t. Case. 1 (mod 4). Let = 4t +1. Assig the labels to the vertices of u 1, u,...,u t ad u as i case 1. Put the label 4 to the vertex u t+1. The use the labels 3 ad 1 i this order alterately to label the vertices u t+, u t+3,..., u 4t+1. The vertex ad edge coditios of the above labellig µ are give by v µ (1) = v µ (3) = t, v µ () = v µ (4) = t + 1 ad e µ (0) = e µ (1) = 4t + 1 respectively. Case 3. (mod 4). Take = 4t +. A 4-prime cordial φ for this case is give below. Assig the labels to the vertices u 1, u,..., u t+1 ad u as i case. The put the label 3 to the vertices u t+ ad u t+3. The the remaiig rim vertices u t+4, u t+5,...,u 4t+ are labelled with 1 ad 3 i this order alterately. Note that the vertex u 4t+ received the label 1. For this labellig, we have the followig vertex ad edge coditios: v φ (1) = t, v φ () = v φ (3) = v φ (4) = t + 1 ad e φ (0) = e φ (1) = 4t +. Case 4. 3 (mod 4) ad 3,7. Let = 4t + 3. We describe a 4-prime cordial labellig ψ as follows: as i case 3, we assig the labels to the vertices u 1, u,..., u t+3 ad u. The the vertices u t+4, u t+5,..., u 4t are labelled with 3 ad 1 i this order alterately. Clearly the label of u 4t is 3. Fially the last three vertices, amely, u 4t+1, u 4t+ ad u 4t+3 are labelled by 1. With referece to the vertex coditio v ψ (1) = v ψ () = v ψ (3) = v ψ (4) = t + 1 ad the edge coditio e ψ (0) = e ψ (1) = 4t + 3, ψ is a 4-prime cordial labellig for this case. From the above four cases, we ca coclude that W is 4-prime cordial for 3,4,7 ad this completes the existece part. Example 1. A 4-prime cordial labellig of W 11 is give i Figure 1. Figure 1: A 4-prime cordial labellig of W 11

5 34 R. Poraj et al. Corollary 1. If 0, 3 (mod 4), 3, 4, 7 the the double coe C + K is 4-prime cordial. Take the vertex ad edge sets as i Theorem 1Let v be the ew vertex joied to the rim vertices of W. Here p = + ad q = 3. We divide the proof ito two cases. Case 1. 0 (mod 4). Let = 4t, t > 1. Assig the label 4 to the vertex v. The assig the remaiig vertices as i Theorem 1.If f deote this labellig the v f (1) = v f (3) = t, v f () = v f (4) = t + 1 ad e f (0) = e f (1) = 6t. Case. 3 (mod 4). Let = 4t + 3, t > 1. Assig the label to the vertex v ad the assig the remaiig vertices as i Theorem 1. If g deote this labellig the v g () = t +, v g (1) = v g (3) = v g (4) = t + 1 ad e g (0) = 6t + 4, e g (1) = 6t + 5. Thus for 3, 4, 7 ad 0,3 (mod 4), C + K is 4-prime cordial. The ext ivestigatio is about the 4-prime cordiality of the gear graph G. Theorem. The gear graph G is 4-prime cordial for all. Cosider a wheel W with vertex ad edge sets are as i Theorem 1. Now subdivide each edge u i u i+1, u u 1 with the vertices v i where i rus from 1 to 1 ad v respectively. We observe that the order ad size of G are +1 ad 3 respectively. We divide the proof ito four cases. Case 1. 0 (mod 4). Let = 4t. We label the cetral vertex u by. Now cosider the rim vertices u i where 1 i. Assig the label to the vertices u 1, u,..., u t. The put the labels 3 ad 1 respectively to the vertices u t+1 ad u t+. Now put the label 3 to all the remaiig vertices u t+3, u t+4,..., u 4t. Next we cosider the vertices v i where 1 i. The first t vertices v 1, v,..., v t are labelled with 4 the for v t+1 ad v t+, we use the labels 3 ad 1 respectively. The remaiig vertices from the set {v t+3,...,v 4t } are labelled by 1. If f deotes the above metioed labelig the we have the followig vertex ad edge coditios: v f (1) = v f (3) = v f (4) = t, v f () = t + 1 ad e f (0) = e f (1) = 6t. Case. 1 (mod 4). Let = 4t + 1. Here, we costruct a 4-prime cordial labellig g as follows: assig the labels to the vertices u, u 1, u,..., u t ad v 1, v,..., v t as i case 1. The put the label 4 to u t+1. Now cosider the remaiig rim vertices u t+, u t+3,..., u 4t+1. All these vertices are labelled with 3. Similarly, we assig the label 1 to all the vertices from the set {v t+1,...,v 4t+1 }. Note that,

6 AAM: Iter. J., Vol. 1, Issue 1 (Jue 017) 35 v g (1) = t, v g () = v g (3) = v g (4) = t + 1 ad e g (0) = 6t + 1, e g (1) = 6t +. Case 3. (mod 4). Take = 4t +. Assig the labels to the vertices u, u 1, u,..., u t ad the label 4 to the vertices v 1, v,..., v t as i case 1. For the vertices u t+1 ad v t+1, we assig the labels, 4 respectively. The ext two rim vertices, amely, u t+, u t+3 are labelled with 3, 1 respectively. The the remaiig rim vertices u t+4, u t+5,..., u 4t+ are labelled by 3. The vertices v t+, v t+3 are assiged with 3, 1 respectively. Fially, the remaiig ulabeled vertices v t+4, v t+5,..., v 4t+ are labelled by 1. If the above metioed labellig is take as φ the v φ (1) = v φ (3) = v φ (4) = t +1, v φ () = t + ad e φ (0) = e φ (1) = 6t + 3. Case 4. 3 (mod 4). Put = 4t + 3. We describe a 4-prime cordial labellig ψ as follows: as i case 1, assig the labels to the vertices u, u 1, u,..., u t ad v 1, v,..., v t. The we cosider the remaiig rim vertices u t+1, u t+,..., u 4t+3. First we put the labels, 4 to the vertices u t+1, u t+ respectively. The remaiig rim vertices are labelled by 1. For the vertices v t+1, v t+,..., v 4t+3, we use the label 4 to v t+1 ad the the rest vertices v t+, v t+3,..., v 4t+3 are labelled with 3. It is easy to verify the vertex ad edge coditios of the above labellig ψ are v ψ (1) = t + 1, v ψ () = v ψ (3) = v ψ (4) = t + ad e ψ (0) = 6t + 4, e ψ (1) = 6t + 5. Hece, all gears are 4-prime cordial. Next we prove that, for all values of, the helm H is 4-prime cordial. Also we discuss the 4-prime cordiality of closed helm CH. Theorem 3. The helm H is 4-prime cordial for all. Cosider the wheel W with vertex ad edge sets as i Theorem 1. Let v 1, v,..., v be the pedet vertices adjacet to u 1, u,..., u respectively. Clearly p = + 1 ad q = 3. We costruct four differet types of labellig so that H is 4-prime cordial. For this, we split the proof ito four cases. Case 1. 0 (mod 4). Let = 4t. We preset a 4-prime cordial labellig f as follows: first we cosider the rim vertices u i where 1 i. The vertices u 1, u,..., u t are labelled with the iteger. The remaiig rim vertices u t+1, u t+,..., u 4t are labelled alterately with 3 ad 1 i this order so that u 4t received the label 1. For the cetral vertex, we use the label. Now, we cosider the pedet vertices v i where 1 i. Assig the label 4 to the pedet vertex v i, 1 i t i which v i is a eighbour of the vertex u i such that the label of u i is. Assig the labels 3, 1 to the vertices v t+1, v t+ respectively. The remaiig pedet vertices v i from v t+3 to v 4t are labelled with 1 ad 3 accordig as i is odd or eve. For this vertex labellig, we have the vertex coditio as v f (1) = v f (3) = v f (4) = t, v f () = t + 1 ad e f (0) = e f (1) = 6t.

7 36 R. Poraj et al. Case. 1 (mod 4). Let = 4t + 1. I this case, we fix the labels to the vertices u, u i, 1 i t ad v i, 1 i t as i case 1. Cosider the rim vertices u i, t + 1 i 4t + 1. Put the umber 4 to u t+1. The remaiig vertices are labelled with 3 or 1 accordig as i is eve or odd. Now, the oly ulabeled vertices are v i, t + 1 i 4t + 1. Assig the labels 3, 3, 1 to the vertices v t+1, v t+, v t+3 respectively. Fially, the vertices v i, t + 4 i 4t + 1 are labeled with 1 or 3 accordig as its eighbour u i, t + 4 i 4t + 1 received the label 3 or 1. Let g deote the above metioed labellig. It is easy to check that v g (1) = t, v g () = v g (3) = v g (4) = t + 1 ad e g (0) = 6t +, e g (1) = 6t + 1. Case 3. (mod 4). Let = 4t +. Here, we assig the labels to the vertices u, u i, v i, 1 i t ad u i, v i, t + 4 i 4t + 1 as i case. Next we assig the labels 3, 1 to the vertices u 4t+, v 4t+ respectively. The assig the labels to the vertices u t+1, u t+, u t+3 respectively with, 3, 1 ad v t+1, v t+, v t+3 with 4, 3, 1 respectively. The above labellig φ satisfies the vertex coditio v φ (1) = v φ (3) = v φ (4) = t + 1, v φ () = t + ad the edge coditio e φ (0) = e φ (1) = 6t + 3. Case 4. 3 (mod 4). Put = 4t + 3. I this case, we assig the labels to the vertices u, u i, v i, 1 i t + 1 are as i case 3. Now cosider the remaiig rim vertices. Assig the labels 3, 1, 3 to the vertices u t+, u t+3, u t+4 respectively. The the vertices u i, t + 5 i 4t + 3 are labeled with 3 or 1 accordig as i is odd or eve. For the pedet vertices v t+,..., v 4t+3, we first allocate the labels 3, 3, 1 to the vertices v t+, v t+3, v t+4 respectively. The rest of the pedet vertices are labeled with 1 or 3 accordig as i is odd or eve. Let ψ be the above said labellig the it is easy to verify that v ψ (1) = t + 1, v ψ () = v ψ (3) = v ψ (4) = t + ad e ψ (0) = 6t + 5, e ψ (1) = 6t + 4. From Figure, it is easy to see that H 3 is 4-prime cordial. Figure : A 4-prime cordial labelig of H 3 Hece, the helm H is 4-prime cordial for all values of. Corollary. For every values of, the closed helm CH is 4-prime cordial. Clearly, the order ad size of CH are + 1, 4 respectively. We divide the proof ito four

8 AAM: Iter. J., Vol. 1, Issue 1 (Jue 017) 37 cases. Case 1. 0 (mod 4). For = 4, the Figure 3 shows that CH 4 is 4-prime cordial. Figure 3: A picture of a gull. Whe > 4, let = 4t ad we assig the labels to the vertices as i case 1 of Theorem 3. The relabel the vertices u 4t 1, v 4t 1 with 1, 3 respectively. Case. 1 (mod 4). Let = 4t + 1. Assig the labels to the vertices as i case of Theorem 3. Case 3. (mod 4). For = 6, the Figure 4 shows that CH 6 is 4-prime cordial. Figure 4: A 4-prime cordial labelig of CH 6 Whe > 6, let = 4t +, t > 1 ad assig the labels to the vertices as i case 3 of Theorem 3. The relabel the vertices u 4t+, v 4t+ with 1, 3 respectively. Case 4. 3 (mod 4). A 4-prime cordial labellig of CH 3 is give i Figure 5. For > 3, let = 4t + 3, t > 0 ad assig the labels to the vertices as i case 4 of Theorem 3.

9 38 R. Poraj et al. Figure 5: A 4-prime cordial labelig of CH 3 Note that i each case, the umber of edges labelled with 1 ad ot labelled with 1 are, respectively. Hece, the closed helm CH is 4-prime cordial. Next we examie the butterfly graph. Theorem 4. The butterfly graph By m, is 4-prime cordial. Let u 1, u,..., u be the vertices of the first copy of the cycle C ad v 1, v,..., v be that of the secod copy. By symmetry, without loss of geerality, we ca uify u 1 ad v 1. Let w 1, w,..., w m be the pedet vertices of By m,. Now, we depict a 4-prime cordial µ as follows: assig the first copy of the vertices of the cycle with the labels, 4 such that each label assiged to exactly vertices i ay order. For the secod copy, sice v 1 = u 1 is already labelled by, we ca focus the remaiig vertices v,..., v. These vertices are labelled by 3 ad 1 alterately i this order. If we do like this, oe ca easily check that the label of v is 3. Also, at this phase, each labels are used exactly times except the label 1 which is used oe less that of the others i umber. Now we shift ito the pedat vertices. Here, we have four cases. Case 1. m 0 (mod 4). Let m = 4t, t 1. Here, we use each labels exactly t times to label the vertices w 1, w,..., w m i ay order. Case. m 1 (mod 4). Let m = 4t + 1, t 0. I this case, assig the labels to the vertices w 1, w,..., w m 1 as i case 1. The put the iteger 1 to w m. Case 3. m (mod 4). Let m = 4t +, t 0. I this case, assig the labels to the vertices w 1, w,..., w m 1 as i case

10 AAM: Iter. J., Vol. 1, Issue 1 (Jue 017) 39. The put the iteger to w m. Case 4. m 3 (mod 4). Let m = 4t +3, t 0. Here, assig the labels to the vertices w 1, w,..., w m 1 as i case. The put the iteger 1 to w m. We ca refer the Table 1 for the existece of a 4-prime cordial labellig µ of By m,. Table 1. Vertex ad edge coditios of 4-prime cordial labellig µ of By m, Values of v µ (1) v µ () v µ (3) v µ (4) e µ (0) e µ (1) m 0 (mod 4) 1 + t + t m 1 (mod 4) + t + t + 1 m (mod 4) t t + 1 m 3 (mod 4) t t + Hece, By m, is a prime cordial graph. Fially we look ito the friedship graph. Theorem 5. The friedship graph C (t) 3 is 4-prime cordial if ad oly if t 1. Let V (C (t) 3 ) = {u,u i,v i : 1 i t} ad E(C (t) 3 ) = {uu i,u i v i,v i u : 1 i t}. Obviously the order ad size of this graph are respectively t +1 ad 3t. Assig the label to the vertex u. The we cosider the followig cases to label the vertices u i, v i. Case 1. t 0 (mod ). Let t = r, r > 0. Assig the label to the vertices u 1, u,..., u r. The put the label 3 to the vertices u r+1, u r+,..., u r. Similarly, we allocate the iteger 4 to the vertices v 1, v,..., v r. The remaiig ulabeled vertices are labelled by 1. If φ deotes the above metioed labellig the we have v φ (1) = v φ (3) = v φ (4) = r, v φ () = r + 1 ad e φ (0) = e φ (1) = 3r. Case. t 1 (mod ). Let t = r + 1, r > 0. Assig the labels to the vertices u i, v i : 1 i r, as i case 1. The put the labels 3, 4 to the vertices u r+1, v r+1. If f deotes the above metioed labellig the we have v f (1) = v f (4) = r, v f () = v f (3) = r + 1 ad e f (0) = 3r + 1, e f (1) = 3r +. Thus the friedship graph C (t) 3 is 4-prime cordial if ad oly if t Coclusio Graph labelig plays a importat role of various fields of sciece ad few of them are astroomy,

11 40 R. Poraj et al. codig theory, x-ray crystallography, radar, circuit desig, commuicatio etwork addressig, database maagemet, secret sharig schemes, ad models for costrait programmig over fiite domais. So study of graph labelig is a essetial oe. I this paper, we have studied the 4-prime cordiality of wheel, gear, double coe, helm, butterfly graph ad closed helm. Ivestigatio of 4-prime cordiality of grid, caterpillars, drago, prism are ope problems for future researchers. Ackowledgmets The authors wish to thak the aoymous referees for their careful readig ad costructive commets o earlier versio of this article, which resulted i better presetatio of this article. REFERENCES Cahit, I. (1987), Cordial Graphs: A weaker versio of Graceful ad Harmoious graphs, Ars Combi., Vol. 3, pp Gallia, J. A. (015), A dyamic survey of graph labellig, The Electroic Joural of Combiatorics, Vol. 18, pp. #Ds6. Graham, R. L. ad Sloae N. J. A. (1980), O additive bases ad harmoious graphs, SIAM J. Alg. Discrete Math, Vol. 1, pp Harary, F. (001), Graph theory, Narosa Publishig house, New Delhi. Kwog, H., Lee, S. M. ad Ng, H. K. (010), O product-cordial idex sets of cyliders, Cogr. Numer., Vol. 06, pp Kwog, H., Lee, S. M. ad Ng, H. K. (01), O product-cordial idex sets ad friedly idex sets of -regular graphs ad geeralized wheels, Acta Math. Siica, (Egl. Ser.), Vol. 8, No. 4, pp Poraj, R., Rajpalsigh ad Sathish Narayaa, S. (016), k-prime cordial graphs, J. Appl. Math. & Iformatics, Vol. 34, No.3-4, pp Rosa, A. (1967), O certai valuatios of the vertices of a graph, Theory of Graphs (Iterat. Symposium, Rome, July 1966), Gordo ad Breach, N. Y. ad Duod Paris, pp Salehi, E. (010), PC-labellig of a graph ad its PC-sets, Bull. Ist. Combi. Appl., Vol. 58, pp Salehi, E. ad Mukhi, Y. (01), Product cordial sets of log grids, Ars Combi., Vol. 107, pp Seoud, M.A. ad Salim, M.A. (010), Two upper bouds of prime cordial graphs, JCMCC, Vol. 75, pp Shiu, W. C. (013), Product-cordial idex set for Cartesia product of a graph with a path, J. Combi. Number Th., Vol. 4, No. 3, pp Sudaram, M., Poraj, R. ad Somasudaram, S. (004), Product cordial labellig of graphs, Bull. Pure ad Appl. Sci. (Math. & Stat.), Vol. 3E, pp Sudaram, M., Poraj, R. ad Somasudaram, S. (005), Prime cordial labelig of graphs, J. Idia Acad. Math., Vol. 7, pp