Ratio Mathematica Volume 44, 2022 Totally magic d-lucky number of graphs N. Mohamed Rilwan * A. Nilofer † Abstract In this paper we introduce a new labeling named as, totally magic d-lucky labeling, find the totally magic d-lucky number of some standard graphs like wheel, cycle, bigraph etc. and find the totally magic d-lucky number of some zero divisor graphs. A totally magic d-lucky labeling of a graph G = (V, E) is a labeling of vertices and label the graph's edges using the total label of its incident vertices in such a way that for any two different incident vertices u and v, their colors dgu, dtv= Nvtu+ dg v are distinct and for any different edges in a graph, their weights are same Where represents the degree of u in a graph and N represents the open neighbourhood of u in a graph. Keywords: Totally magic d-lucky labeling, totally magic d-lucky number, zero divisor graphs. 2010 AMS subject classification: 05C78 ‡ * Assistant Professor, Department of Mathematics, Sadakathullah Appa College (Autonomus), Rahmath nagar, Tirnelveli-627011, India; e-mail- rilwan2020@gmail.com. † Research scholar, register number 18221192092017, Research center, Department of Mathematics, Sadakathullah Appa College (Autonomus), Rahmath nagar, Tirunelveli 627011, Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli 627012, India; e-mail- lourdhunilofer@gmail.com ‡ Received on June 9 th , 2022. Accepted on Sep 1st, 2022. Published on Nov 30th, 2022. doi: 10.23755/rm.v44i0.920. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 316 N. Mohamed rilwan, A. Nilofer 1. Introduction In [2], The idea of lucky labeling was first proposed by Czerwinski, Grytczuk, and Zelezny. In [1], The idea of "d-lucky labeling" was developed by Indira Rajasingh, D. Ahima Emilet, and D. Azhubha Jemilet. [1] Let l: V(G) → N is a vertex labeling. If for each pair of incident vertices of u and v, c(u) ≠ c(v) holds where c(u) = , c(v) = , represents the degree of u and N(u) represents the open neighbourhood of the vertex u in a graph, then the labeling l is a d- lucky labeling. A graph's d-lucky number is the smallest value of labeling required to label the graph. Motivated by this labeling, we introduce Totally magic d-lucky labeling. A graph's total labeling is a mapping from the union of the vertex set and the edge set to positive integers. If the sum of the edge label and the label of the edge's end points has the same constant, the total labeling is said to be totally magic labeling. In [5,] we learned about the totally magic labeling. A totally magic d-lucky labeling of a graph G = (V, E) is a labeling of vertices and label the edges of the graph by the sum of the labels of its incident vertices in such a way that for any two different incident vertices u and v, their colors , are distinct and for any different edges in a graph, their weights are same Where represents the degree of u in a graph and N represents the open neighborhood of u in a graph. 2. Totally magic d-lucky labeling In this section we introduce a new labeling named as the totally magic d-lucky labeling and apply it on the cycle, path, complete graph, bigraph, and wheel. Definition 2.1 Define t: V(G) → {1, 2, …, p}and label the edges of E(G) as the label of the edge's incident vertices added together. The labeling is said to be Totally magic d- lucky labeling if and t(u) + t(v) + t(uv) 0 (mod 2) where u, v V(G) and . The totally magic d-lucky number of G, tdln(G) is defined as the lowest value of p for which the graph G has totally magic d-lucky labeling. Theorem 2.2. For a cycle graph Cn, tdln (Cn) = Proof. Let G be the cycle graph. Let V(G) = {vi: 1≤ i ≤ n} and E(G) = {vi vi+1: 1≤ i ≤ n-1} ᴜ {vnv1} Case. (i). When . Let t: V(Cn)→{1,2,...,p}defined by for 1≤i≤n-1 t(vi)=i for 1≤ i ≤ n Then the induced edge labelling is, t(vivi+1) =2i+1 for 1≤ i ≤ n-1 317 Totally magic d-lucky number of graphs t(vnv1) = n+1 We observe that, dt(v1) = n+4 dt(vi) = 2i+2, for 2≤i≤n-1 dt(vn) = n+2 dt(vi) ≠ dt(vi+1) and dt(v1) ≠ dt(vn) t(vi)+ t(vi+1) + t(vivi+1)=4i+2 ≡ 0 (mod 2) for 2≤ i≤ n-1 and t(v1)+ t(vn)+t(v1vn) = 2n+2≡ 0(mod2) case (ii). When n 0 (mod 2) Define t: V(Cn) → {1,2, …, p}as follows, for t(vi) = Then the induced edge labelling is, t(vivi+1) = 3 for 2 ≤ i ≤ n-1 t(vnv1) = 3 we observe that, dt(vi) = 6 if i is odd dt(vi) = 4 if i is even dt(vi) ≠ dt(vi+1) and t(vi)+ t(vi+1) + t(vivi+1)= 6 0 (mod 2) It can be easily verified that weights of the incident vertices are pair wise distinct and have the common totally magic d-lucky constant for its edges. Thus, the totally magic d-lucky number of cycle graph is 2. ■ Theorem 2.3 Every path Pn has tdln(Pn)=2 Proof Let Pn be the path graph, V(Pn)={vi: } and E(Pn) ={vivi+1: for } Define t: V(Pn) → {1,2,…,p}as follows: t(vi) = Then the induced edge labelling is, t(vi vi+1) =3 for all edges in Pn we observe that, when n is even, dt(v1) = 3 dt(vi) = 4 if i ≡ 0(mod 2) dt(vi) = 6 if i ≡ 1(mod 2) dt(vn) = 2 dt(vi) ≠ dt(vi+1) for all i and t(vi) + t(vi+1) + t(vivi+1) = 6 ≡ 0(mod 2). When n is odd dt(v1) = 3 = dt(vn) 318 N. Mohamed rilwan, A. Nilofer dt(vi) = 4 if i ≡ 0(mod 2) dt(vi) = 6 if i ≡ 1 (mod 2) dt(vi) ≠ dt(vi+1) for all i and t(vi) + t(vi+1) + t(vivi+1) = 6 ≡ 0 (mod 2). Hence tdln (Pn) = 2. ■ Theorem 2.4 For a complete graph kn, tdln (kn) = n Proof In complete graph Kn, Each and every pair of vertices are close together. Define t:V(Kn)→{1,2,…,p}as follows: t(vi) = i : Then the induced edge labelling is, t(vivj) = i+j for all edges in Pn we observe that, for 1 ≤ i ≤ n dt(vi) = dt(vi) ≠ dt(vj) and t(vi) + t(vj) + t(vivj) = 2(i+j) ≡ 0(mod 2). tdln (Kn) = n. It is simple to confirm that the colors of the pair wise incident vertices are distinct and that the sum of the labels for each edge and the incident vertices of its edges is even. ■ Theorem 2.5 For a bigraph Km, n, tdln (Km, n) = 1. Proof A bigraph's vertices can be divided into two separate subsets, V1 and V2, and each edge of the bigraph connects a point on each subset. Km, n indicates a bigraph. Let V (Km, n) = V1 V2 where V1 = { } and V2 = and E (Km, n) = { }. Define t: V (Km, n) → {1, 2, …, p} as follows: t(ui) = 1, t(vj) =1 Then the induced edge labeling is t(uivj) = 2 for all edges in Km, n We observe that, , , t(ui) + t(vj) + t(uivj) =2(i+j) ≡ 0 (mod 2). It is obvious that all incident vertices have pair wise different colors and that all of the edges in the Km, n graph have the same totally magic d-lucky constant. Hence tdln (Km, n) =1. ■ Theorem 2.6 For a wheel graph Wn, tdln (Wn) = . Proof A wheel graph is obtained by joining a vertex to all the vertices of a cycle graph. It is denoted by Wn for n>3, where n is the number of vertices in the graph. Let V(Wn) = { } and E(Wn) = { } Case(i) When n 1(mod2) 319 Totally magic d-lucky number of graphs Define a labeling t:V(Wn) → {1,2,…,p}as follows: t( ) = 1, t( ) = i-1 for Then the induced edge labelling is t( ) = i for We observe that, dt(u1) = dt( ) = n+5 dt( ) = 2i+2 for 3≤ i ≤n-1 dt(un) = n+3 dt(u1) ≠ dt(ui) for 2 dt(ui) ≠ dt(ui+1) for dt(un) ≠ dt(u1) t(u1)+t(ui)+t(u1ui) = 2i ≡ 0 (mod 2); t(ui)+t(ui+1)+t(uiui+1) = 4i-2 ≡ 0 (mod 2), for 2≤ i ≤ n-1; t(u1)+t(un)+t(un) = 2n ≡ 0(mod 2) Hence tdln (Wn) = n-1 Case (ii) When n ≡ 0 (mod 2) Define t:V(Wn)→{1,2,…,n} as follows: t(ui) = i for Then the induced edge labelling is ; = 2i+1 -1 t(unu2)=n+2 we observe that, dt( ) = dt( ) = n+7 dt(ui) = 2i+4 for 3≤ i ≤ n-1 dt(un) = n+5 dt ≠ dt(ui) for 2 dt(ui) ≠ dt(ui+1) for dt(un) ≠ dt( ) for t( ) + t(ui+1) + t( ui+1 ) = 2i+2 ≡ 0 (mod 2), t(ui) + t(ui+1) + t(uiui+1) = 4i+2 ≡ 0 (mod 2), t( ) + t(un) + t( un) = 2n+4 ≡ 0 (mod 2). Hence tdln (Wn)= n It is simple to confirm that all incident vertices' colors are pairwise different and preserve the totally magic d-lucky constant for all of the graph's edges Wn. ■ 320 N. Mohamed rilwan, A. Nilofer 3. Totally magic d-lucky number of some zero divisor graphs In this part, the totally magic d-lucky number of some zero divisor graphs is examined. Theorem. 3.1 For R = Zk , k = mn, m=2,3 and n>3 be a prime number, tdln(Γ(R)) = 1 Proof Consider G0=Γ(R) where R=Zk, k=mn Case(i) when m=2,n >3be a prime. By the definition of zero divisor graph, Assume V(G0) = {2, 4... 2 (n-1), n} = {vi: 1 ≤ i≤ n}, E(G0) = {vi vn: vi V (Γ (R))-{vn}}. We have dg(vi) = m-1, dg(vn) = n-1, 1 ≤ i ≤ 2(n-1) Define t:V(G0)→{1,2,…,p} as follows: t(vi) = 1for 1≤i≤n Then the induced edge labeling is, t(e) = 2 for all edges e in G0 we observe that, dt(vi) = 2m-2, dt(vn) = 2n-2 dt(vi)≠ dt(vn) for and t(vi)+ t(vn) + t(vivn) = 4 ≡ 0 (mod 2) Hence tdln(G0) = 1 Case(ii) when m=3, n>3 be a prime. In this graph, we have V(G0) = V1(G0) V2(G0) where V1(G0) = {n,2n}, V2(G0) = {3i: 1≤i≤n-1} and E(G0) = {uv: u V1(G0), v V2(G0)}. Hence dg(u) = n-1 for all u V1(G0), dg(v) = m-1, for all v V2(G0) and |E(G0)| = 2n-2 Define a labeling t: V(G0)→{1,2,…,p} as follows: t(u) = 1 for all u V1(G0) t(v) = 1 for all v V2(G0) Then the induced edge labeling is, t(uv) = 2 for all uv E(G0) We observe that, dt(u) = 2n-2, dt(v) = 2m-2 dt(u) dt(v) for all u V1 , for all v V2 and t(u) + t(v) + t(uvi) = 4 ≡ 0(mod 2) for all uv E(G0) Hence tdln(G0) = 1. ■ Theorem 3.2 For R = Zk, k =m 2 n, n >3 be a prime number, tdln(Γ(R)) = Proof Assume G0 = Γ(R) 321 Totally magic d-lucky number of graphs Case(i) When m=2, In this case(G0) has partitioned into two sets V1(G0),V2(G0). V1(G0) contains the multiples of n in Zk, V2(G0) contains the multiples of m excluding 2n in Zk. Let V1(G0) = { } and V2(G0) = { } |V(G0)| =2n+1 E(G0) = |E(G0)| = 4n-4 Hence dg(ri) = n-1, for i {1,3}; dg( ) = 2n-2; dg(sj) = m+1, sj ; dg(sj) = m-1, sj V2(G0) – {4, 8, …, 4n-4} Define a labeling t: V(G0) → {1, 2, …, p} as follows: t(ri) = 1 for 1 ≤ i ≤ 3; t(sj) = 1 for 1 ≤ j ≤ 2n-2; Then the induced edge labelling is, t(risj) = 2 for all risj E(G0) We observe that, dt(ri) = 2n-2, i {1,3}, dt( ) = 4n-4, dt(sj) = 2m-2, sj V(Γ(R))-{4,8,…,4n-4}, dt(sj) = 2m+2, sj {4,8,…,4n-4} dt( ) ≠ dt(sj) for all V1(G0) , sj` V2(G0) and t(ri) + t(sj) + t (risj) = 4 ≡ 0 (mod 2) for all edges in G0 Hence tdln(G0)=1. Case(ii) when m = 3, In this case, the vertex set of G0 partitioned into two sets V1 and V2. WhereV1 = {n, 2n, 3n, 4n, 5n, 6n, 7n, 8n} = {u1, u2, u3, u4, u5, u6, u7, u8}and V2 = {3, 6, 9, …, 9n-3} - {n, 2n} = {v1, v2, v3, …, v3n-1}. E(G0) = {uivi: for all ui V1 , vi {9,18,27,…,9(n-1)} {uivi : ui {3n,6n} vi V2 } {u3,u6}. Hence dg(ui) =n-1 for all ui V1-{3n,6n}; dg(ui) = 3n-2, i={3,6}; dg(vi) = 8 for all vi {9,18,…,9(n-1)}; dg(vi) = 2, vi V2-{9,18,…,9(n-1)} . Define the labelling t V(G0)→ {1,2,…,p} as follows: t(ui) = 1, for ,ui V1; t(u6) = 2, u6 V1; t(vi) = 1, for all vi V2. Then the induced edge labellings are, t(uivi) = 2 for all uivi E(G0) t(u3u6) = 3 t(u6vi) = 3 for all vi V2 We observe that, dt(ui) = 2n-2, 322 N. Mohamed rilwan, A. Nilofer dt( ) = 6n-3, dt( ) = 6n-4, dt(vi) = 5 for all vi V2(G0)-{9,8,…,9(n-1)} dt(vi) = 17, vi {9,8,…,9(n-1)} dt(ui) dt(vi), dt( ) dt( ) and t(ui)+t(vi)+t(uivi) = 4 ≡ 0 (mod 2) for all edges in G0 t(u3) + t(u6) + t(u3u6) = 6 ≡ 0 (mod 2) t(u6) + t(vi) + t(u6vi) = 3 for all vi V2 It can be easily verified that weights of all the incident vertices are distinct and all the edges of the graph have common totally magic d-lucky constant. Hence tdln (G0) = 2. ■ Theorem 3.3 Let R = be a commutative ring with unity. For the zero-divisor graph Γ(R), tdln(Γ(R)) = M-1 where M = (m1, m2, m3, …, mk), mi’s are distinct prime numbers, ni’s are positive integers. Proof Consider G0 = Γ(R) be a zero-divisor graph of commutative ring R = where mi’s are prime numbers and ni’s are positive integers. The vertex set of G0 consists of different blocks, V(G0) = where ( , ) (0, 0…, 0) and ( , ) ( ). = {( , ): =0 if = and | and ∤ if {0,1,2,…,ni-1}} All the vertices in are adjacent to all the vertices in if for all i= 1,2,…,k. The vertices in form a clique in G0 if for all i= 1,2,..,k Hence we have, for each u , dg(u) = if is clique ; dg(u) = -1 + if is not a clique. Define a labeling t: V(G0) {1, 2, …, p}as follows, Label the vertices of the block as 1 if the block is not form a clique. If the block is form a clique, label the vertices of clique uj as , where , Then the induced edge labellings are, if the block is not form a clique, t(e) = 2 for all edge e in this block if the block is form a clique, t(ujuj+1) = 2j+1 for all , t(uqu1) = q+1 323 Totally magic d-lucky number of graphs Let T = Max ( ) if form a clique We observe that, for each u , we have, , dt(u) = t(u)+t(v)+t(uv) ≡ 0 (mod 2) for all uv E(G0) Hence tdln(G0) = T= M-1 where M=Max( ). It can be easily verified that colors of all the incident vertices are pairwise distinct and have the common constant for all the edges of the our given graph. ■ 4. Conclusions In this paper, we introduced a new labeling, totally magic d-lucky labeling, found the totally magic d-lucky number of some standard graphs and some zero divisor graphs. In future, we use this labeling in some other graphs. 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