Mean cordiality of some snake graphs

Transcription

1 Palestie Joural of Mathematics Vol. 4() (015), Palestie Polytechic Uiversity-PPU 015 Mea cordiality of some sake graphs R. Poraj ad S. Sathish Narayaa Commuicated by Ayma Badawi MSC 010 Classificatios: 05C78. Keywords ad phrases: Triagular sake, alterate triagular sake, double triagular sake. Abstract Let f be a fuctio from the vertex set V (G) to {0, 1, }. assig the label f(u)+f(v) For each edge uv. f is called a mea cordial labelig if v f (j) 1 ad e f (j) 1, i,j {0, 1, }, where v f (x) ad e f (x) respectively are deote the umber of vertices ad edges labeled with x (x = 0, 1, ). A graph with a mea cordial labelig is called a mea cordial graph. I this paper we ivestigate mea cordial labelig behavior of double triagular sake, alterate triagular sake, double alterate triagular sake. 1 Itroductio All graphs i this paper are fiite, udirected ad simple. The vertex set ad edge set of a graphg are deoted by V (G) ad E(G) respectively. Let p, q deotes the umber of vertices ad edges i G. Poraj et al. defied the mea cordial labelig of a graph i []. Mea cordial labelig behavior of path, cycle, star, complete graph, wheel, comb, mg, P m P, P, triagular sake etc have bee ivestigated i [, 6]. Also, Albert william et al. [1] have studied about the mea cordial labelig behaviour of certai graphs like subdivisio of a bistar S(B m, ), particular type of caterpillar, Baaa tree ad path baaa tree. Here we ivestigate the mea cordial labelig behavior of double triagular sake, alterate triagular sake, double alterate triagular sake. The symbol x stads for smallest iteger greater tha or equal to x. Terms ad defiitios ot defied here are used i the sese of Harary [4]. Prelimiary Results I this sectio we write some basic defiitios ad results which are eeded for the ext sectio. Defiitio.1. Letf be a fuctio fromv (G) to{0, 1, }. For each edgeuv ofgassig the label. f is called a mea cordial labelig if v f (j) 1 ad e f (j) 1, f(u)+f(v) i,j {0, 1, }, where v f (x) ad e f (x) deote the umber of vertices ad edges labeled with x (x = 0, 1, ) respectively. A graph with a mea cordial labelig is called a mea cordial graph. Defiitio.. The triagular sake T is obtaied from the path P by replacig each edge of the path by a triagle C. Defiitio.. A alterate triagular sake A(T ) is obtaied from a path u 1 u...u by joiig u i ad u i+1 (alteratively) to ew vertexv i. That is every alterate edge of a path is replaced by C. Defiitio.4. A double alterate triagular sake DA(T ) cosists of two alterate triagular sakes that have a commo path. That is, a double alterate triagular sake is obtaied from a path u 1 u...u by joiig u i ad u i+1 (alteratively) to two ew vertices v i ad w i. Defiitio.5. A double triagular sake D(T ) cosists of two triagular sakes that have a commo path. Theorem.6. [5] The triagular sake T ( > 1) is mea cordial iff 0 (mod ). Mai Results Theorem.1. Mea cordial labelig behaviour of Alterate triagular sake A(T ) is give below:

2 440 R. Poraj ad S. Sathish Narayaa a. Mea cordial if the triagle starts from u ad eds with u. b. Mea cordial if the triagle starts from u 1, eds with u ad 0 (mod ). c. Not Mea cordial if the triagle starts from u 1, eds with u ad 1, (mod ). d. Mea cordial if the triagle starts from u, eds with u. Proof. Case a. The triagle starts from u ad eds with u. I this case p =, q =. Sub case 1. 1 (mod ). Let = t+1, t > 1. Assig the label 0 to t+1 path vertices u 1,u,...,u t+1. The assig to the ext t path vertices; assig 1 to the remaiig path vertices. The we move to the vertices of degree. Label the vertices v 1,v,...,v t by 0. The assig the label to the vertices v t 1,v t,...,v t. Fially assig the label 1 to the vertices v t,v t 1,...,v t 5. The above vertex labelig f, satisfies the mea cordial coditio by table 1. i Table 1. Whe t = 1, the correspodig mea cordial labelig of A(T 4 ) is give i figure Figure.1 Sub case. (mod ). Let = t+. Assig the label 0 to the vertices of the first t triagles ad the 1 to the ext t triagles. The assig the label to the vertices of the remaiig t triagles. Fially assig the label 0, to the pedet vertices u 1 ad u respectively. I this case the vertex ad edge coditio is give i table. i Table. 4 Sub case. 0 (mod ). Let = t. Assig the labels to the vertices as i sub case the relabel the verticesu 1 by ad the vertex u t+1, a vertex of t th triagle by. The table shows that the above vertex labelig f is a mea cordial labelig. i 0 1 Table. For the cases b & c, p = ad q =. Case b. The triagle starts from u 1, eds with u ad 0 (mod ). Let = t. Assig the label 0 to the vertices of the first t triagles. The 1 to the vertices of the ext t triagles. Fially assig the label to the vertices of the remaiig t triagles. The table 4 establishes that the above vertex labelig f, satisfies the mea cordiality coditio. Case c. The triagle starts from u 1, eds with u ad 1, (mod ).

3 Mea cordiality of some sake graphs 441 i 0 1 Table 4. Sub case 1. 1 (mod ). Suppose f is a mea cordial labelig, the v f (0) = v f (1) = v f () =. This forces the maximum value of e f (0) is 1. That is ef (0) 1. Sice the size of A(T ) is, f ca ot be satisfies the edge coditio of the mea cordial labelig. Sub case. (mod ). Suppose f is a mea cordial labelig, the v f (0) = v f (1) = v f () =. I this case e f(0) 4, a cotradictio. Case d. The triagle starts from u, eds with u. Here p =, q =. Sub case 1. 0 (mod ). Let = t, t > 1. Assig the label 0 to the firstt+1 vertcesu 1,u,...,u t+1 of the path. The label the ext t 1 vertices u t+,u t+,...,u t by 1 ad assig the label to the remaiig vertices of the path. Now we move to the vertices with degree. Assig the label 0 to the first t 1 vertices ad the the ext t+1 vertices receives the label 1. Fially assig the label to the remaiig t 1 vertices. From the table 5, we ca coclude that the above vertex labelig, say f, is a mea cordial labelig. i Table 5. Whe t = 1, the mea cordial labelig of A(T ) is give i figure Figure. Sub case. 1 (mod ). Let = t+1. Assig the label 0 to the first t triagles ad the label the vertices of the ext t triagles by 1. The assig the label to the vertices of the remaiig t triagles. Fially assig the label 0 to the vertex u 1. The vertex ad edge coditios of the above labelig f is give i table 6. i Table 6. Sub case. (mod ). Let = t +. Cosider the path vertices u 1,u,...,u. Assig the label 0 to the vertices u 1,u,...,u t+1 ad label the ext t + 1 vertices u t+,u t+,...,u t+ by 1. The assig the label to the vertices u t+,u t+4,...,u t+. The we move to the vertices of degree. These are labeled i the followig patter. w 1 w... w t 1 w t w t+1... w t w t w t 1... w t

4 44 R. Poraj ad S. Sathish Narayaa Fially assig the label 0 to the pedet vertex. I this case the followig table 7 shows that the above vertex labelig f, is a mea cordial labelig. i Table 7. A mea cordial labelig ofa(t 1 ) with the coditio that the triagle starts fromu, eds with u is give i figure Figure. Theorem.. Mea cordial labelig behaviour of double alterate triagular sake DA(T ) is give below: a. DA(T ) is mea cordial if the triagle starts fromu ad eds withu ad 0, (mod ). b. Not mea cordial if the triagle starts from u, eds with u ad 1 (mod ). c. Mea cordial if the triagle starts from u 1, eds with u, >. I this case DA(T ) is ot mea cordial. d. Mea cordial if the triagle starts from u, eds with u ad 1 (mod ). e. Not mea cordial if the triagle starts from u, eds with u ad 0, (mod ). Proof. For the cases a & b, p = ad q = 5. Case a. The triagles starts from u ad eds with u ad 0, (mod ). Sub case 1. 0 (mod ). oidet Assig the label to the vertices of the first t double triagles by 0, ext t double triagles by 1 ad the last t double triagles by. The replace the label of the vertexu t+1 by. Fially assig the label to the pedet vertices. The labelig f give i above is mea cordial from table 8. i 0 1 Table 8. Sub case. (mod ). Label the vertices of DA(T ) as i subcase 1 ad assig the label to the vertices of the last double triagles. The replace the label of the vertex u t+1 by 0. The vertex coditio ad edge coditio of the labeligs f is show i table 9. i Table 9. Case b. The triagles starts from u, eds with u ad 1(mod ). Suppose f is a mea cordial labelig. The v f (0) = v f (1) = v f () =. But e f (0) 1. This is a cotradictio. 5

5 Mea cordiality of some sake graphs 44 Case c. The triagles starts from u 1, eds with u. I this case p = ad q =. Cosider the graph DA(T ). Suppose f is a mea cordial labelig. The we have two cases. v f (0) = or v f (0) = 1. If v f (0) = 1 the e f (0) = 0, a cotradictio. Suppose v f (0) =. Note that the label 0 should be assiged to the adjacet vertices (otherwise e f (0) = 0). The e f (0) = e f () = 1, e f (1) = or e f (0) = 1, e f (1) = 4, e f () = 0, a cotradictio. Therefore DA(T ) is ot mea cordial. Sub case 1. 0 (mod ). Assig the label to the vertices of the first t double triagles by 0. Put the label 1 to the vertices of the ext t triagles. Fially assig the label to the vertices of the last t double triagles. The table 10 shows that f is a mea cordial labelig. i 0 1 Table 10. Sub case. 1 (mod ). Assig the label 0 to the first t+1 path vertices the assig the label 1 to the ext t vertices of the path. The remaiig t vertices of the path are labeled by. The vertices v i ad w i are labeled as give below. v 1 v... v t v t+1 v t+... v t v t+1 v t+... v t w 1 w... w t 1 w t w t+1... w t 1 w t w t+1... w t The values of ad are give i table 11 i Table 11. Sub case. (mod ). Assig the label 0 to the vertices of first t+ double triagles the assig the label to the ext t vertices of the double triagles by 1 ad the last t triagles by. Fially replace the labels of the vertices u t+, wt+, u t+, w t+1 by 1, 1, 1, respectively. The followig table 1 shows that the above vertex labelig f is a mea cordial labelig. i Table 1. For the cases d & e, p = ad q =. Case d. The triagle starts from u, eds with u ad 1 (mod ). Assig the label 0 to the vertices of the first t double triagles, 1 to the vertices of the ext t double triagles ad to the vertices of the last t double triagles. Put the label 0 to the pedet vertex u 1. Table 1 establish that the labelig f give above is a mea cordial labelig. Case e. The triagle starts from u, eds with u. Sub case 1. 0 (mod ).

6 444 R. Poraj ad S. Sathish Narayaa i Table 1. Suppose f is a mea cordial labelig. I this case, either v f (0) = e f (0), a cotradictio. Sub case. (mod ). or. I both cases Here v f (0) =. But e f (0). This cotradictio proves that there does ot exists a mea cordial labelig. A mea cordial labelig of DA(T 10 ) with the coditio that the triagles starts from u 1, eds with u is give i figure Figure.4 Theorem.. The double triagular sake D(T ) is mea cordial iff >. Proof. Case 1. =. Follows from case c of theorem.. Case. =. I this case, v f (0) = or. If v f (0) = the e f (0) 1 which is ot possible. If v f (0) = the 0 should be labeled to the vertices of ay triagle. Otherwise the value ofe f (0) lower tha. I the case of 0 labeled i the vertices of the triagle, the value e f () is ot greater tha. This is a cotradictio to the size. Hece D(T ) is ot mea cordial. Case. >. Sub case 1. 1 (mod ). Let = t + 1. Assig the label 0 to v i (1 i t), 1 to v t+i (1 i t) ad to v t+i (1 i t+1). Label the vertices u i (1 i t+1) by 0, u t+1+i (1 i t) by 1 ad u t+1+i by. The we move to the vertex w i. Assig the labels to w i as i v i. The followig table 14 shows that the above labelig f is a mea cordial labelig. i Table 14. Sub case. 0 (mod ). Let = t, t > 1. Assig the label 0 to the vertices u i (1 i t+1), v i (1 i t) ad w i (1 i t 1). Put the label 1 to the vertices u j (t+ j t), v j (t+1 j t) ad w j (t j t 1). Fially assig the label to the vertices u r (t + 1 r t), v r (t+1 r t 1) ad w r (t r t 1). The table 15 give below shows that the above labelig f is a mea cordial labelig. 5 5 i Table 15. 5

7 Mea cordiality of some sake graphs 445 Sub case. (mod ). Let = t+. First we cosider the path vertices. Assig the label 0 to u i (1 i t+1), 1 to u j (t + j t + ) ad to u r (t + r t). Now we move to the vertices v i (1 i ). The first t+1 vertices are labeled by 0 ad the vertices v i (t+ i t + 1) are labeled by 1 the the last t vertices of v i are labeled by. The vertex labelig of w i (1 i ) is give below. w 1 w... w t w t+1 w t+... w t w t+1 w t+... w t Let f be the labelig defied above. The values ad where i = 0, 1, give i table 16 proves that f is a mea cordial labelig. i Table Refereces [1] Albert william, Idra rajasigh ad s. Roy1, Mea Cordial Labelig of Certai Graphs, J. Comp. & Math. Sci., 4 (4), (01). [] I. Cahit, Cordial Graphs: A weaker versio of Graceful ad Harmoious graphs, Ars combi., (1987), [] J. A. Gallia, A Dyamic survey of Graph labelig, The Electroic joural of Combiatorics, 18 (011), # DS6. [4] F. Harary, Graph theory, Addisio wesley, New Delhi. [5] R. Poraj, M. Sivkumar ad M. Sudaram, Mea cordial labelig of Graphs, Ope Joural of Discreate Mathematics, Vol., No. 4, 01, [6] R. Poraj ad M. Sivakumar, O Mea cordial graphs, Iteratioal Joural of Mathematical combiatorics, (01), Author iformatio R. Poraj ad S. Sathish Narayaa, Departmet of Mathematics, Sri Paramakalyai College, Alwarkurichi , Tamil Nadu, Idia. ÔÓÒÖ Ñ Ø Ñ ÐºÓÑ Ø ÖÚ Ñ ÐºÓÑ