Treating subjects as vertex graceful graphs, vertex graceful labeling, caterpillar, actinia graphs, Smarandachely vertex m-labeling

Balaganesan, P.

Renuka, J.

Selvaraju, P.

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NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

Vertex Graceful Labeling-Some Path Related Graphs

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International J.Math. Combin. Vol.3(2013), 44-49Vertex Graceful Labeling-Some Path Related GraphsP.Selvaraju1, P.Balaganesan2 and J.Renuka31Department of Mathematics, Vel Tech Engineering College, Avadi, Chennai- 600 062, Tamil Nadu, India2Department of Mathematics, School of Engineering, Saveetha University, Chennai- 602 105, Tamil Nadu, India3 Departments of Mathematics, Sai Ram College of Engineering, Chennai - 600 044, IndiaE-mail: pselvarr@gmail.com, balki2507@yahoo.co.inAbstract: In this article, we show that an algorithm for VG of a caterpillar and provedthat A(mj, n) is vertex graceful if mj is monotonically increasing, 2 ≤ j ≤ n, when n is odd,1 ≤ m2 ≤ 3 and m1 < m2, (mj , n) ∪ P3 is vertex graceful if mj is monotonically increasing,2 ≤ j ≤ n, when n is odd, 1 ≤ m2 ≤ 3, m1 < m2 and Cn ∪ Cn+1 is vertex graceful if andonly if n ≥ 4.Key Words: Vertex graceful graphs, vertex graceful labeling, caterpillar, actinia graphs,Smarandachely vertex m-labeling.AMS(2010): 05C78§1. IntroductionA graph G with p vertices and q edges is said to be vertex graceful if a labeling f : V (G) →{1, 2, 3 · · ·p} exists in such a way that the induced labeling f+ : E(G) → Zq defined byf+((u, v)) = f(u) + f(v)(mod q) is a bisection. The concept of vertex graceful (V G) wasintroduced by Lee, Pan and Tsai in 2005. Generally, if replacing q by an integer m andfS : E(G)→ Zm also is a bijection, such a labeling is called a Smarandachely vertex m-labeling.Thus a vertex graceful labeling is in fact a Smarandachely vertex q-labeling.All graphs in this paper are finite simple graphs with no loops or multiple edges. Thesymbols V (G) and E(G) denote the vertex set and edge set of the graph G. The cardinality ofthe vertex set is called the order of G. The cardinality of the edge set is called the size of G. Agraph with p vertices and q edges is called a (p, q) graph.§2. Main ResultsAlgorithm 2.11. Let v1, v2 · · · vn be the vertices of a path in the caterpillar. (refer Figure 1).2. Let vij be the vertices, which are adjacent to vi for 1 ≤ i ≤ n and for any j.3. Draw the caterpillar as a bipartite graph in two partite sets denoted as Left (L) which1Received April 10, 2013, Accepted August 15, 2013.Vertex Graceful Labeling-Some Path Related Graphs 45contains v1, v2j , v3, v4j , · · · and for any j and Right (R) which contains v1j , v2, v3j , v4, · · · andfor any j. (refer Figure 2).4. Let the number of vertices in L be x.5. Number the vertices in L starting from top down to bottom consecutively as 1, 2, · · · , x.6. Number the vertices in R starting from top down to bottom consecutively as (x +1), · · · , q. Note that these numbers are the vertex labels.7. Compute the edge labels by adding them modulo q.8. The resulting labeling is vertex graceful labeling.(16)v42v41v32v21v14v12v11(5)v43(6)(4)(14)(13)(9) (10) (11) (20)(2)(1)v1 v2(12)v3(3)v4(15)v5(7)10 11 12v31171413 151651 52v 53v 54vv(17)55v(19)(18)180 19 4 5 6 7 823(8)v11Figure 1: A caterpillarv21v3v41v42v43v5v11v12v13v14v2v31v32v4v51v52v54v55v1(3)(2)(1)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(17)(18)v53(19)(20)(16)121110131415161718902341Figure 2: A caterpillar as bipartite graphDefinition 2.2 The graph A(m,n) obtained by attaching m pendent edges to the vertices ofthe cycle Cn is called Actinia graph.Theorem 2.3 A graph A(mj , n),mj is monotonically increasing with difference one, 2 ≤ j ≤ nis vertex graceful, 1 ≤ m2 ≤ 3 when n is odd.Proof Let the graph G = A(mj , n),mj be monotonically increasing with difference one,2 ≤ j ≤ n, n be odd with p = n + mn(mn+12 ) −m1(m1+12 ),m1 = m2 − 1 vertices and q = pedges. Let v1, v2, v3, · · · , vn be the vertices of the cycle Cn. Let vij(j = 1, 2, 3, · · · , n) denotethe vertices which are adjacent to vi. By definition of vertex graceful labeling, the required46 P.Selvaraju, P.Balaganesan and J.Renukavertices labeling arevi =(i− 1)2(m2 +(i+1)2)+ 1, 1 ≤ i ≤ n, i is odd,(m2 + 1)(n+1)2 +(n−12)2+ (i−2)2(m2 +i2)+ i2 , 1 ≤ i ≤ n, i iseven.vij =(n− 1)2(m2 +(n+1)2)+ i−12(m2 +i−32)+ i+12 + j, 1 ≤ j ≤ m2 + i− 1, i is odd;(i− 2)2(m2 +i−22)+ i2 + j, 1 ≤ i ≤ m2 + i− 1, i is even.The corresponding edge set labels are as follows:Let A = {ei = vivi+1/1 ≤ i ≤ n− 1 ∪ en = vnv1}, whereei =[(m2 + 1)(n+ 1)2+(n− 12)2+m2(i− 1) + i(i+ 1)2+ 1](mod q)for 1 ≤ i ≤ n. B = {eij = vivij/1 ≤ i ≤ n}, whereeij =[(n− 1)2(m2 +(n+ 1)2)+ (i− 1)(m2 +i− 12)+(i+ 1)2+ j + 1](mod q)for 1 ≤ i ≤ n and i is odd, j = 1, 2, · · · ,m2 + i− 1. C = {eij = vivij/1 ≤ i ≤ n}, whereeij =[(m2 + 1)(n+ 1)2+(n− 12)2+i− 22(2m2 + i− 1) + i+ j](mod q)for 1 ≤ i ≤ n and i is even, j = 1, 2, · · · ,m2 + i− 1.Hence, the induced edge labels of G are q distinct integers. Therefore, the graph G =A(mj , n) is vertex graceful for n is odd, and m ≥ 1. 2Theorem 2.4 A graph A(mj , n) ∪ P3,mj be monotonically increasing, 2 ≤ j ≤ n is vertexgraceful, 1 ≤ m2 ≤ 3, n is odd.Proof Let the graph G = A(mj , n) ∪ P3,mj be monotonically increasing ,2 ≤ j ≤ n ,n is odd with p = n + 3 + mn(mn+1)2 − m1 (m1+1)2 ,m1 < m2 vertices and q = p − 1 edges.Letv1, v2, v3, · · · , vn be the vertices of the cycle Cn. Let vij(j = 1, 2, 3, · · · , n) denote thevertices which are adjacent to vi. Let u1, u2, u3 be the vertices of the path P3. By definition ofvertex graceful labeling, the required vertices labeling arevi =i−12(m2 +i+12)+ 1; 1 ≤ i ≤ n, i is odd;(m2 + 1)(n+1)2 +(n−12)2+ (i−2)2(m2 +i2)+ i2 + 2; 1 ≤ i ≤ n, i is even.vij =n−12(m2 +n+12)+ i−12(m2 +i−32)+ i+12 + j + 2; 1 ≤ i ≤ n, i is odd,i− 22(m2 +i−22)+ i2 + j + 2; 1 ≤ i ≤ n, i is even.ui =n−12(m2 +n+12)+ i+12 for i = 1, 3 and u2 = p.The corresponding edge labels are as follows:Let A = {ei = vivi+1/1 ≤ i ≤ n− 1 ∪ en = vnv1}, whereei =[(m2 + 1)(n+ 1)2+(n− 12)2+m2(i− 1) + i(i+ 1)2+ 3](mod q)Vertex Graceful Labeling-Some Path Related Graphs 47for 1 ≤ i ≤ n. B = {eij = vivij/1 ≤ i ≤ n}, whereeij =[(n− 1)2(m2 +(n+ 1)2)+ (i− 1)(m2 +i− 12)+(i+ 1)2+ j + 3](mod q)for 1 ≤ i ≤ n and i is odd, j = 1, 2, · · · ,m2 + i− 1. C = {eij = vivij/1 ≤ i ≤ n}, whereeij =[(m2 + 1)(n+ 1)2+(n− 12)2+i− 22(2m2 + i− 1) + i+ j + 2](mod q)for1 ≤ i ≤ n and i is even, j = 1, 2, · · · ,m2 + i− 1. D = {ei = uiui+1 for i = 1, 2}, whereei =[n− 12(m2 +n+ 12+ i+ 1](mod q)for i = 1, 2. Hence, the induced edge labels of G are q distinct integers. Therefore, the graphG = A(mj , n) ∪ P3 is vertex graceful for n is odd. 2Definition 2.5 A regular lobster is defined by each vertex in a path is adjacent to the path P2.Theorem 2.6 A regular lobster is vertex graceful.Proof Let G be a 1- regular lobster with 3n vertices and q = 3n − 1 edges. Letv1, v2, v3, · · · , vn be the vertices of a path Pn . Let vi be the vertices, which are adjacentto vii1 and vii1 adjacent to vii2 for 1 ≤ i ≤ n and n is even .The theorem is proved by two cases.By definition of Vertex graceful labeling, the required vertices labeling areCase 1 n is evenvi =3i−12 ; 1 ≤ i ≤ n, i is odd,3(n+ i)2; 1 ≤ i ≤ n, i is even.vi1 =3(n+ i)− 12/ 1 ≤ i ≤ n, i is odd3i− 2)2+ 3/ 1 ≤ i ≤ n, i is even.vi2 =3(i−1)2 + 2; 1 ≤ i ≤ n, i is odd,3(n+ i)2− 1; 1 ≤ i ≤ n, i is even.The corresponding edge labels are as follows:Let A = {ei = vivi+1/1 ≤ i ≤ n−1}, where ei =(3(n+ 2i)2+ 1)(mod q) for 1 ≤ i ≤ n−1,B = {ei1 = vivi1/1 ≤ i ≤ n}, where ei1 =(3(n+ 2i)2− 1)(mod q) for 1 ≤ i ≤ n and i is odd,C = {ei1 = vivi1/1 ≤ i ≤ n}, where ei1 =(3(n+ 2i)2)(mod q) for 1 ≤ i ≤ n and is even,D = {ei2 = vi1vi2/1 ≤ i ≤ n}, where ei2 =(3(n+ 2i)2)(mod q) for 1 ≤ i ≤ n and i is odd,E = {ei2 = vi1vi2/1 ≤ i ≤ n}, where ei2 =(3(n+ 2i)2− 1)(mod q) for 1 ≤ i ≤ n and iseven.48 P.Selvaraju, P.Balaganesan and J.RenukaCase 2 n is oddvi =3i− 12; 1 ≤ i ≤ n, i is odd,3(n+ i) + 12; 1 ≤ i ≤ n, i is even,vi1 =3(n+i)2 ; 1 ≤ i ≤ n, i is odd,3(i− 2)2+ 3; 1 ≤ i ≤ n, i is even,vi2 =3(i−1)2 + 2; 1 ≤ i ≤ n, i is odd,3(n+ i− 1)2+ 1; 1 ≤ i ≤ n, i is even.The corresponding edge labels are determined by A = {ei = vivi+1/1 ≤ i ≤ n − 1},where ei =(3(n+ 2i+ 1)2)(mod q) for 1 ≤ i ≤ n − 1, B = {ei1 = vivi1/1 ≤ i ≤ n}, whereei1 =(3(n+ 2i)− 12)(mod q) for 1 ≤ i ≤ n and i is odd, C = {ei1 = vivi1/1 ≤ i ≤ n}, whereei1 =(3(n+ 2i) + 12)(mod q) for 1 ≤ i ≤ n and is even, D = {ei2 = vi1vi2/1 ≤ i ≤ n}, whereei2 =(3(n+ 2i) + 12)(mod q) for 1 ≤ i ≤ n and i is odd, E = {ei2 = vi1vi2/1 ≤ i ≤ n}, whereei2 =(3(n+ 2i)− 12)(mod q) for 1 ≤ i ≤ n and is even. Hence the induced edge labels of Gare q distinct edges. Therefore, the graph G is vertex graceful. 2Theorem 2.7 Cn ∪ Cn+1 is vertex graceful if and only if n ≥ 4.Proof Let G = Cn ∪ Cn+1 with p = 2n+ 1 vertices and q = 2n+ 1 edges. Suppose thatthe vertices of the cycle Cn run consecutively u1, u2, · · · , un with un joined to u1 and that thevertices of the cycle Cn+1 run consecutively v1, v2, · · · , vn+1 with vn+1 joined to v1.By definition of vertex graceful labeling(a) u1 = 1, un = 2, ui = 2i for i = 2, 3, · · · , ⌊(n + 1)/2⌋, uj = 2(n − j) + 3 for j =⌊(n+ 3)/2⌋, · · · , n− 1.(b) v1 = 2, v2 = 2n− 1 and(i) v3s+t = 2n−4t−6s+7, t= 0, 1, 2, s = 1, 2, · · · , ⌊(n+1−3t)/6⌋ if s = ⌊n+ 1− 3t6⌋ < 1then no s.(ii) Write α(0) = 0, α(1) = 4, α(2) = 2, β(0) = 0, β(1) = 3 = β(2)vn+1−3s−t = 2n− 6s− α(t), t = 0, 1, 2, s = 0, 1, · · · , ⌊n− 5− β(t)6⌋. If s = ⌊n− 5− β(t)6⌋ < 0then no s value exists.(iii) We consider as that vi to f(i); and suppose that n− 2 = θ mod(3), 0 ≤ θ ≤ 2. Thereare 2 + θ vertices as yet unlabeled. These middle vertices are labeled according to congruenceclass of modulo 6.Vertex Graceful Labeling-Some Path Related Graphs 49Congruence classn = 0 (mod 6 )f((n + 2)/2) = n + 2, f((n + 4)/2) = n + 3,f((n + 6)/2) = n + 4n = 1 (mod 6 )f((n + 1)/2) = n + 2, f((n + 3)/2) = n + 3,f((n + 5)/2) = n + 4, f((n + 7)/2) = n+5n = 2 (mod 6 ) f((n + 2)/2) = n + 2, f((n + 4)/2) = n + 3n = 2 (mod 6 )f((n + 1)/2) = n+4, f((n+3)/2) = n+3,f((n+5)/2) = n+2n = 4 (mod 6 )f((n + 2)/2) = n+5, f((n+3)/2) = n+4,f((n+4)/2) = n+3, f((n+5)/2) = n+2n = 4 (mod 6 ) f((n + 3)/2) = n+3, f((n+5)/2) = n+2To check that f is vertex graceful is very tedious. But we can give basic idea. The Cncycle has edges with labels {2k+2/k = 4, 5, · · · , n−1}∪{0, 3, 5, 7}. In this case all the labelingof the edges of the cycle Cn+1 run consecutively v1v2 as follows:1, (2n−1, 2n−3), (2n−11, 2n−13, 2n−15), · · · , (2n+1−12k, 2n−1−12k, 2n−3−12k), · · · ,middle labels,· · · , (2n+3−12k, (2n+5−12k, (2n+7−12k), · · · , (2n−21, 2n−19, 2n−17), (2n−9, 2n − 7, 2n − 5), 2. The middle labels depend on the congruence class modulo and are bestsummarized in the following table. If n is small the terms in brackets alone occur.Congruence classn = 0(mod6) · · · (11, 9), 6, 4, 7, (13, 15, 17) · · ·n = 1(mod6) · · · (13, 11), 6, 4, 7, (13, 15, 17) · · ·n = 2(mod6) · · · (11), 6, 4, 7, (9) · · ·n = 2(mod6) · · · (13), 7, 4, 6, (9, 11) · · ·n = 4(mod6) · · · (15, 9), 6, 4, 7(11, 13) · · ·n = 4(mod6) · · · (9), 7, 6, 4(11, 13, 15) · · ·Thus, all these edge labelings are distinct. 2References[1] J.A.Gallian, A Dynamic Survey of graph labeling, The Electronic journal of Coimbino-torics, 18 (2011), #DS6.[2] Harary F., Graph Theory, Addison Wesley, Mass Reading, 1972.[]3 Sin-Min Lee, Y.C.Pan and Ming-Chen Tsai, On vertex- graceful (p,p+1) Graphs, Congres-sus Numerantium, 172 (2005), 65-78.[4] M.A Seoud and A.E.I Abd el Maqsoud, Harmonious graphs, Utilitas Mathematica, 47(1995), pp. 225-233.[5] P.Balaganesan, P.Selvaraju, J.Renuka,V.Balaji, On vertex graceful labeling, Bulletin ofKerala Mathematics Association, Vol.9,(June 2012), 179-184.

Vertex Graceful Labeling-Some Path Related Graphs

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