Ratio Mathematica Volume 47, 2023 Minimal Reinhard Zumkeller divisor cordial graphs A. Ruby Priscilla * S. Firthous Fatima † Abstract In this paper, the notion of minimal Reinhard Zumkeller divisor cordial labeling has been introduced. Let G = (V, E) be a simple graph and γ : V (G) → minimum {2i × 3, 2j+1 × 5, 2k+1 × 7, 2l × 3 × 5, 2m × 3 × 7 where i, j, k, l, m ≥ 1} be an injection such that the sum of the cardinality of exponent of γ(V (G)) should be equal to the order of the graph G. For each edge uv, assign the label 1 if γ(u)|γ(v) or γ(v)|γ(u) where γ(u) and γ(v) are Zumkeller numbers and the label 0 if γ(u) ∤ γ(v) and also if |eγ|(0) − eγ|(1)| ≤ 1 then γ is called minimal Reinhard Zumkeller divisor cordial labeling. This paper elucidates how the Zumkeller number, which is the generalization of the perfect number, goes along with the divisibility concept of the number theory and the cordial labeling technique. It also probes the existence of minimal Reinhard Zumkeller divisor cordial labeling of path, cycle, star K1,s, complete bipartite, complete graph Kn for n < 17 , tadpole graph Tn,k for all values of n and k. Keywords: Zumkeller graph, divisor cordial labeling, Zumkeller divisor cordial graph. 2010 AMS subject classifications: 05C78. 1 *Research Scholar, Reg.No:18221192092003, Department of Mathematics, Sadakathullah Appa College (Autonomous), Affiliated to Manonmaniam Sundaranar University, Tirunelveli, Tamilnadu, India and Assistant Professor, Department of Mathematics, Sarah Tucker College (Autonomous), Tirunelveli-7. ruby@sarahtuckercollege.edu.in †Assistant Professor, Department of Mathematics, Sadakathullah Appa College (Autonomous), Rahmath Nagar, Tirunelveli-627011, kitherali@yahoo.co.in 1Received on October 07, 2022. Accepted on June 10, 2023. Published on June 30, 2023. DOI: 10.23755/rm.v39i0.866. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 279 A. Ruby Priscilla and S. Firthous Fatima 1 Introduction Graphs regarded here are finite, undirected and simple. The symbols V (G) and E(G) denote the vertex set and the edge set of a graph G. Most of the graph labeling methods trace their origin to the one introduced by Rosa [1967]. A graph labeling is an assignment of integer to the vertices or edges or both subject to certain conditions. Labelled graph has many branch out applications such as coding theory, missile guidance, X-ray, crystallography analysis, communication network addressing systems, astronomy, radar, circuit design, database management etc., The concept of cordial labeling was introduced by Cahit [1987]. Varatharajan et al. [2011] introduced divisor cordial labeling. If the sum of all the proper positive divisors of a positive integer is equal to the number, then the number is called perfect number. Generalizing the concept of perfect numbers R.H.Zumkeller defined a new type of number as a Zumkeller number. Peng and Rao [2013] established several results and conjectures on Zumkeller numbers. The notion of Zumkeller labeling of some cycle related graphs was investigated by Balamurugan et al. [2014]. Murali et al. [2017] proved results about Zumkeller cordial labeling of cycle related graphs. Shahbaz and Mahmood [2020] proved that Zumkeller number is either a super totient or a hyper totient number. Graph labeling has a potent communication between the number theory and graph network. The idea behind this work fosters us to develop a graph labeling technique called minimal Reinhard Zumkeller divisor cordial labeling by pooling the divisor cordial graph labeling technique and characteristics of Zumkeller number which is one of the engrossing parts of the number theory. Because of the existence of diverse γ vertex labeling design for some graph structure, minimal condition is emphasized. As the work has been focused on minimization condition, Zumkeller number chosen for the concept depicted herein is a sequence of least even Zumkeller numbers. An added reason for not using sequence of odd Zumkeller numbers is mainly due to point up the minimum sequence of Zumkeller numbers. Mahanta et al. [2020] stated that 945 is the smallest odd Zumkeller number. In this paper we discuss the existence of minimal Reinhard Zumkeller divisor cordial labeling of path, cycle, star K1,s, complete bipartite, complete graph Kn for n < 17, tadpole graph Tn,k for all values of n and k. 2 Preliminaries Definition 2.1. Varatharajan et al. [2011] Let G = (V, E) be a simple graph and f : V → {1, 2, . . . |V |} be a bijection. For each edge uv, assign the label 1 if either f(u) | f (v) or f(v) | f(u) and the label 0 if f(u) ∤ f(v). f is called a divisor cordial labeling if |ef (0) − ef (1)| ≤ 1. A graph with a divisor cordial labeling is called a divisor cordial graph. 280 Minimal Reinhard Zumkeller divisor cordial graphs 1. ef|(0)is the number of edges of the graph G having label 0 under f | 2. ef|(1) is the number of edges of the graph G having label 1 under f | Definition 2.2. Peng and Rao [2013] A positive integer n is said to be a Zumkeller number if the positive divisors of n can be partitioned into two disjoint subsets of equal sum. A Zumkeller partition for a Zumkeller number n is a partition {A, B} of the set of positive divisors of n so that each of A and B sums to the same value. Proposition 2.3. Peng and Rao [2013] For any prime p ̸= 2 and positive integer k with p ≤ 2k+1 − 1, the number 2kp is a Zumkeller number. Fact 2.4. Peng and Rao [2013] Let the prime factorization of an even Zumkeller number n be 2kpk11 p k2 2 . . . p km m where k is a positive integer. Then at least one of ki must be odd. Definition 2.5. Balamurugan et al. [2014] A simple graph G = (V, E), where V is vertex set and E is edge set of G is said to admit a Zumkeller labeling if there exists an injective function f : V → N such that f∗ : E → N defined as f∗(xy) = f(x)f(y) is a Zumkeller number for xy ∈ E; x, y ∈ V . The labelled graph G is called as a Zumkeller graph. Definition 2.6. Murali et al. [2017] Let G = (V, E) be a graph. An injective function f : V → N is said to be a Zumkeller cordial labeling of the graph G if there exists an induced function f∗ : E → {0, 1} defined by f∗(xy) = f(x)f(y) satisfies the following conditions 1. For every xy∈E, f∗(xy) = { 1 , f (x) f (y) is a Zumkeller number ; 0 , otherwise 2. |ef∗ (0) − ef∗ (1)| ≤1 Definition 2.7. Murali et al. [2017] A graph G = (V, E) which admits a Zumkeller cordial labeling is called a Zumkeller cordial graph. 3 Main Results The vertex labeling γ mention in the definition 3.1 is defined by using Proposition 2.3 and fact 2.4. Definition 3.1. Let G = (V, E) be a simple graph and γ : V (G) → minimum{ 2i × 3, 2j+1 × 5, 2k+1 × 7, 2l × 3 × 5, 2m × 3 × 7 where i, j, k, l, m ≥ 1 } be an injection satisfying any one of the following conditions |i|+|j| = |V (G)| or |i| + |k| = |V (G)| or |i| + |l| = |V (G)| or |i| + |m| = |V (G)| or |i| + |j| + |k| = 281 A. Ruby Priscilla and S. Firthous Fatima |V (G)| or |i|+|j|+|l| = |V (G)| or |i|+|j|+|m| = |V (G)| or |i|+|j|+|k|+|l| = |V (G)| or |i|+|j|+|k|+|m| = |V (G)| or |i|+|j|+|k|+|l|+|m| = |V (G)| i.e., the sum of the cardinality of exponent of γ(V (G)) should be equal to the order of the graph G. For each edge uv, assign the label 1 if γ(u) | γ(v) or γ(v) | γ(u) where γ(u) and γ(v) are Zumkeller numbers and the label 0 if γ(u) ∤ γ(v) and also if |eγ| (0) − eγ| (1)| ≤ 1 then γ is called minimal Reinhard Zumkeller divisor cordial labeling. A graph with a minimal Reinhard Zumkeller divisor cordial labeling is called a minimal Reinhard Zumkeller divisor cordial graph. Theorem 3.2. The path Pn is a minimal Reinhard Zumkeller divisor cordial when n ≡ 0, 1 (mod 2) Proof. Let v1, v2, . . . .vn be the vertices of the path Pn. Label those consecutive adjacent vertices in the order as 2i × 3 and 2l × 3 × 5 where 1 ≤ i ≤ n 2 and 1 ≤ l ≤ n 2 for the path having even number of vertices and for the path having odd number of vertices 1 ≤ i ≤ n+1 2 and 1 ≤ l ≤ n−1 2 and also |i| + |l| = |V (G)|. If 2i × 3 |2l × 3 × 5 then the consecutive adjacent vertices contribute 1 to each edge and if 2i × 3 ∤ 2l × 3 × 5 then the consecutive adjacent vertices contribute 0 to each edge. Thus eγ|(1) = n 2 and eγ|(0) = n−2 2 if n is even and eγ|(1) = eγ|(0) = n−1 2 if n is odd. Hence |eγ|(0) − eγ|(1)| ≤ 1.Thus Pn is a minimal Reinhard Zumkeller divisor cordial graph. Theorem 3.3. The cycle Cn is a minimal Reinhard Zumkeller divisor cordial when n ≡ 1 (mod 2) , n ≥ 3, n ∈ N Proof. Let v1, v2, . . . .vn be the vertices of the cycle Cn. By making use of the similar pattern described as for path, the cycle of odd order is investigated as a minimal Reinhard Zumkeller divisor cordial graph. Theorem 3.4. The cycle Cn admits a minimal Reinhard Zumkeller divisor cordial when the vertex vn is labelled with 2 × 3 × 7 where n ≡ 0 (mod 2) , n ≥ 4, n ∈ N. Proof. Let v1, v2, . . . .vn be the vertices of the cycle Cn. Label the vertex vn with 2 × 3 × 7 and appertain with the similar pattern described as for path for the remaining vertices results in a minimal Reinhard Zumkeller divisor cordial labeling for the cycle graph. 282 Minimal Reinhard Zumkeller divisor cordial graphs Theorem 3.5. The Wheel graph Wn = K1 + Cn is a minimal Reinhard Zumkeller divisor cordial Proof. Let vo be the center vertex of Wn and label the center vertex as 22 × 5 Case 1. n is even. Label the vertices v1, . . . .vn of Cn as 2i × 3 where 1≤i≤n such that gcd( (20,γ (v1)) , (20,γ (v2)) , . . . . (20,γ (vn)) = 1 and gcd (γ (vg) , γ (vg+1)) > 1 where 1≤g≤n−1 and also |i| + |j| =n+1 =n+1 = |V (Wn)| . We observe that, eγ|(0) =eγ| (1) =n. (1) Case 2. n is odd. Label the center vertex vo as 22×5 and label v1, . . . vn of Cn with the same labeling design mentioned in case 1 Here also, eγ| (0) = eγ| (1) = n (2) Hence, from (1) and (2) we get that |eγ|(0) − eγ|(1)| = { 0 if n is even 0 if n is odd Theorem 3.6. The star graph K1,s is a minimal Reinhard Zumkeller divisor cordial labeling when s ≡ 0, 1(mod 2) Proof. Let G = K1,s be the star graph with vertex set V (G) = {v0 ∪ {vg : 1 ≤ g ≤ s}} where v0is a center vertex and vg’s are pendant vertices and an edge set E(G) = {eg = v0vg : 1 ≤ g ≤ s}. Here we notice that the order of the graph |V (G)| = s + 1 and the size of the graph |E(G)| = s. Case 1. s ≡ 1 (mod 2) Assume γ(v0) = 2 × 3 which is a Zumkeller number. The pendant vertices contribute 1 to its adjacent edges are labelled as follows. γ(vi) = 2 i+1 ×3 for 1≤i≤s−1 2 + 1 and the pendant vertices contributes 0 to its adjacent edges are labelled as follows γ(vj) = 2 j+1×5 for 1≤j≤s−1 2 . And also |i| + |j| = s + 1 Case 2. s ≡ 0 (mod 2) Assume γ(v0) = 2 × 3 which is a Zumkeller number. The pendant vertices contribute 1 to its adjacent edges are labelled as follows. γ(vi) = 2 i+1 ×3 for 1≤i≤s 2 and the pendant vertices contributes 0 to its adjacent edges are labelled as follows γ(vj) = 2 j+1×5 for 1≤i≤s 2 . And also |i| + |j| =s+1 Hence from cases 1 and 2, we get that eγ|(0) = s+1 2 and eγ|(1) = s−1 2 when m is odd and eγ|(0) = eγ|(1) = s 2 when s is even. 283 A. Ruby Priscilla and S. Firthous Fatima Hence ∣∣eγ| (0) − eγ| (1)∣∣ = { 0 if s is even 1 if s is odd Thus ∣∣eγ| (0) − eγ| (1)∣∣ ≤1 . Hence, k1,s is a minimal Reinhard Zumkeller divisor cordial. Theorem 3.7. The complete bipartite graph Kx,z is a minimal Reinhard Zumkeller divisor cordial graph for all values of x, y ≥ 2 Proof. Let V = V1 ∪ V2 be the bipartition of Kx,z such that V1 = {v1, v2, . . . vx} and V2 = {w1, w2, . . . wz} .The order of the complete bipartite graph Kx,z is x + z = f . Case 1. x = z where x and z are even Obviously, there are f 2 vertices in V1 and f 2 vertices in V2. Then label f 4 vertices out of f 2 vertices as 2i×3, where 1≤i≤f 2 and the remaining f 4 vertices get the label as 2j+1×5 , where 1≤j≤f 2 . Label f 4 vertices in V2 as 2i×3 , where f2 +1≤i≤z and the remaining f 4 vertices as 2j+1×5 where f 2 +1≤i≤z . Then the cordiality condition |eγ|(0) − eγ|(1)| = 0 . Case 2. x = z, when x and z are odd. Label f 2 vertices in V1 as follows: Label f 2 +1 2 vertices out of f 2 in V1 as 2i×3, where 1≤i≤ f 2 +1 2 and label the remaining vertices f 2 −( f 2 +1) 2 vertices are labelled as 2i+1×5, where 1≤j≤f 2 − f 2 +1 2 . Then label f 2 vertices in V2 as follows: Label f 2 +1 2 vertices out of f 2 in V2 as 2i × 3, where (f2 +1) 2 + 1 ≤ i ≤ z + 1 and label the remaining vertices z − f 2 +1 2 in V2 as 2j+1 × 5, where f2 − (f2 +1) 2 + 1 ≤ i ≤ z − 1 Then the cordiality condition |eγ|(0) − eγ|(1)| = 1. Case 3. x < z and x + z where z = x + 1, x is odd and z is even There are f+1 2 − 1 vertices in V1 and f+12 vertices in V2 .Label f+1 2 − 1 vertices in V1 as follows: Label f+1 2 2 vertices out of f+1 2 − 1 in V1 as 2i × 3, where 1 ≤ i ≤ f+1 2 2 and the remaining vertices f+1 2 − 1 − f+1 2 2 are labelled as 2j+1 × 5, where 1 ≤ j ≤ f+1 2 − 1 − f+1 2 2 .Label f+1 2 vertices in V2 as follows: Label f+1 2 2 vertices as 2i × 3 , where f+1 2 2 + 1 ≤ i ≤ z and label the left over vertices f+1 2 − f+1 2 2 as 2j+1 × 5, where f+1 2 − f+1 2 2 ≤ j ≤ z − 1 . Following the labeling pattern results in |eγ|(0) − eγ|(1)| = 1 Case 4. x z and x = z + 2. Obviously, there are f 2 +1 vertices in V1 and f 2 −1 vertices in V2. Then label f 2 +1 2 vertices out of f 2 + 1 vertices as 2i × 3 , where 1 ≤ i ≤ f 2 +1 2 and the remaining f 2 +1 2 vertices get the label as 2j+1 × 5, where 1 ≤ j ≤ f 2 +1 2 . Likewise label f 2 − 2 vertices out of f 2 − 1 in V2 as 2i × 3, where f 2 +1 2 + 1 ≤ i ≤ z + 1 and the remaining f 2 −1 − f 2 −2 vertices as 2j+1 × 5 where f 2 +1 2 + 1 ≤ j ≤ z + 1. Then the cordiality condition ∣∣eγ| (0) − eγ| (1)∣∣ = 0. Proceeding like this for all values of x and z, the cordiality condition is satisfied. Hence the complete bipartite is a minimal Reinhard Zumkeller divisor cordial graph. Theorem 3.8. The tadpole Tn, k is a minimal Reinhard Zumkeller divisor cordial graph for all values of n and k Proof. Let v1, v2, . . . , vn be the vertices of cycle Cn and w1, w2, . . . , wk be the vertices of the path Pk. Let Tn,k be the repercussion graph obtained by recognizing a vertex of cycle Cn to an end vertex of the path Pk .Then the order of Tn,k graph is |V (Tn,k)| = n+k and the size of Tn,k graph is |E (Tn,k)| = n+k. Concatenate the pendant vertex of Pk to one of the vertices of Cn with an edge in such a way that   vn+3 2 =w1 for n ≡ 1 (mod 2) if n+3 2 is even vn+3 2 −1=w1 for n ≡ 1 (mod 2) if n+3 2 is odd vn+2 2 = w1 for n ≡ 0 (mod 2) if n+22 is even vn+2 2 − 1 = w1 for n ≡ 0 (mod 2) if n+22 is odd . We contemplate the following cases.Let v1, v2, . . . , vn be the vertices of cycle Cnbe labelled as follows: Case 1. n ≡ 1 (mod 2) and K = 1. Let v1, v2, . . . , vn be the vertices of cycle Cnbe labelled as follows: γ(vg) = { 2i × 3 where g ≡ 1 (mod 2) and 1 ≤ i ≤ n+1 2 2l × 3 × 5 where g ≡ 0 (mod 2) and 1 ≤ l ≤ n+1 2 − 1 (3) Let the vertices of the path Ph w1, w2 be labelled as follows. γ ( vn+3 2 ) = γ(w1) (4) 285 A. Ruby Priscilla and S. Firthous Fatima γ(w2) = 2 n+1 2 +1 × 3 (5) In regards to the labeling designs (3),(4),(5), we get that e|γ(0) = n+K2 ; e | γ(1) = n+K 2 . Hence ∣∣∣e|γ(0) − e|γ(1)∣∣∣ = 0. Case 2. n ≡ 1 (mod 2) and k ≥ 2, where k is even γ(vg) = { 2i × 3 where g ≡ 1 (mod 2) and 1 ≤ i ≤ n+1 2 2l × 3 × 5 where g ≡ 0 (mod 2) and 1 ≤ l ≤ n+1 2 − 1 (6) γ(wh) = { 2i × 3 where h ≡ 1 (mod 2) , 3 ≤ h ≤ k + 1 andn+1 2 + 1 ≤ i ≤ n+1+k 2 2l × 3 × 5 where h ≡ 0 (mod 2) , 2 ≤ h ≤ k andn+1 2 ≤ l ≤ n+1+k 2 − 1 (7) Hence from (6) and (7), we get that e|γ(1) = n+k+12 ; e | γ(0) = n+k−1 2 . Hence ∣∣∣e|γ(0) − e|γ(1)∣∣∣ = 1. Case 3. n ≡ 0 (mod 2) and k ≥ 1 where k is odd γ(vg) = { 2i × 3 where g ≡ 1 (mod 2) and 1 ≤ i ≤ n 2 2l × 3 × 5 where g ≡ 0 (mod 2) and 1 ≤ l ≤ n 2 (8) Let the vertices of the path Ph, w1, w2, . . . , wh be labelled as follows, γ(wh) = { 2i × 3 where h ≡ 0 (mod 2) , 2 ≤ h ≤ k + 1 and n 2 + 1 ≤ i ≤ n+k+1 2 2l × 3 × 5 where h ≡ 1 (mod 2) , 3 ≤ h ≤ k and n 2 + 1 ≤ l ≤ n+k+1 2 − 1 (9) In regards to the above labeling design (8) and (9), we get that e|γ(1) = n+k−12 ; e | γ(0) = n+k+1 2 Hence ∣∣∣e|γ(0) − e|γ(1)∣∣∣ = 1. Case 4. n ≡ 1 (mod 2) where k ≥ 1 where k is odd γ(vg) = { 2i × 3 where g ≡ 1 (mod 2) and 1 ≤ i ≤ n+1 2 2l × 3 × 5 where g ≡ 0 (mod 2) and 1 ≤ l ≤ n+1 2 − 1 (10) γ(wh) = { 2i × 3 where h ≡ 1 (mod 2) , 3 ≤ h ≤ k and n+1 2 + 1 ≤ i ≤ n+k 2 2l × 3 × 5 where h ≡ 0 (mod 2) , 2 ≤ h ≤ k − 1 and n+1 2 + 1 ≤ l ≤ n+k 2 − 1 (11) γ(wk+1) = 2 2 × 5 (12) In regards to the labeling design (10), (11) and (12), we get that e|γ (1) = n+k2 ; e | γ (0) = n+k 2 Hence ∣∣∣e|γ (0) − e|γ (1)∣∣∣ = 0 286 Minimal Reinhard Zumkeller divisor cordial graphs Case 5. n ≡ 0 (mod 2) where K ≥ 2 and K is even γ(vg) = { 2i × 3 where g ≡ 1 (mod 2) and 1 ≤ i ≤ n 2 2l × 3 × 5 where g ≡ 0 (mod 2) and 1 ≤ l ≤ n 2 (13) γ(wh) = { 2i × 3 where h ≡ 0 (mod 2) , 2 ≤ h ≤ k and n 2 + 1 ≤ i ≤ n+k 2 2l × 3 × 5 where h ≡ 1 (mod 2) , 3 ≤ h ≤ k − 1 and n 2 + 1 ≤ l ≤ n+k 2 − 1 (14) γ(wk+1) = 2 2 × 5 (15) In regards to the labeling design (13), (14) and (15), we get that e|γ (1) = n+k2 ; e | γ (0) = n+k 2 Hence ∣∣∣e|γ (0) − e|γ (1)∣∣∣ = 0 Hence from all cases we get that the tadpole Tn,k is a minimal Reinhard Zumkeller divisor cordial graph for all values of n and k. Theorem 3.9. The complete graph Kn is a minimal Reinhard Zumkeller divisor cordial if n ≤ 16 Proof. Obviously K1, K2, K3 are minimal Reinhard Zumkeller divisor cordial graph. The following table 1 brings forth a minimal Reinhard Zumkeller divisor cordial labeling of Kn for 4 ≤ n < 17 Order of Vertex labels Cordiality Condition Kn 4 6,12,20,24 ∣∣eγ| (0) − eγ| (1)∣∣ = 0 5 6,12,28,30,60 ∣∣eγ| (0) − eγ| (1)∣∣ = 0 6 6,12,24,28,30,60 ∣∣eγ| (0) − eγ| (1)∣∣ = 1 7 6,12,24,28,30,48,60 ∣∣eγ| (0) − eγ| (1)∣∣ = 1 8 6,12,24,28,30,48,60,96 ∣∣eγ| (0) − eγ| (1)∣∣ = 0 9 6,12,24,28,30,48,60,96,168 ∣∣eγ| (0) − eγ| (1)∣∣ = 0 10 6,12,24,28,30,48,60,96,168,192 ∣∣eγ| (0) − eγ| (1)∣∣ = 1 11 6,12,24,28,30,48,60,84,96,168,192 ∣∣eγ| (0) − eγ| (1)∣∣ = 1 12 6,12,24,28,30,48,60,84,96,120,168,192 ∣∣eγ| (0) − eγ| (1)∣∣ = 0 13 6,12,24,28,30,48,60,84,96,120,168,192,384 ∣∣eγ| (0) − eγ| (1)∣∣ = 1 14 6,12,24,28,30,48,60,84,96,120,168,192,240,384 ∣∣eγ| (0) − eγ| (1)∣∣ = 1 15 6,12,24,28,30,48,60,84,96,120,168,192,240,384,768 ∣∣eγ| (0) − eγ| (1)∣∣ = 1 16 6,12,24,28,30,48,60,84,96,120,168,192,240,336,384,768 ∣∣eγ| (0) − eγ| (1)∣∣ = 0 Table 1: Minimal Reinhard Zumkeller divisor cordial labeling of Knfor 4 ≤ n < 17 K17 is not Reinhard Zumkeller divisor cordial graph. Since by following the labeling pattern of K16 the vertex labels to be selected for the vertex v17 must be 287 A. Ruby Priscilla and S. Firthous Fatima anyone of them: 480 or 1536 or 672 or by choosing 20 instead of 28 from the above labeling pattern then the cordiality condition is ∣∣eγ| (1) − eγ| (0)∣∣ = 69 − 67 = 2 and ∣∣eγ| (0) − eγ| (1)∣∣ = 69 − 67 = 2 respectively. Since the labeling pattern for each complete graph Kn follows the labeling pattern of its predecessor, for all higher order complete graphs the cordiality condition increases by 1 for each n ≥ 17. 4 Discussion For the notion of minimal Reinhard Zumkeller divisor cordial labeling ,this effort has produced several fresh findings. In order to create a minimal Reinhard Zumkeller divisor cordial graph and introduce a new element to the labeling pattern of various graph structures, the traits of the Zumkeller number are unified with the divisor cordial graph labeling technique. The results that are established in this paper are amalgamated and motivated us to get into the conclusion that for every connected minimal Reinhard Zumkeller divisor cordial graph G, γ(u) ≡ 0(mod6) for some vertex u ∈ V (G) . Deriving similar results for other graph families is an open problem . 5 Conclusions In the present investigation, minimal Reinhard Zumkeller divisor cordial labeling has been introduced and probed for the existence of Reinhard Zumkeller divisor cordial labeling of path, cycle, star K1,s, complete graph Kn for n < 17, complete bipartite and tadpole graph Tn,k for all values of n and k. In future research work, we will develop findings to construct dense minimal Reinhard Zumkeller divisor cordial graphs, book graphs with polygonal pages, generalized Petersen graphs, wheel graphs and product related graphs. Acknowledgement The authors are thankful to the anonymous referees for putting forth their valuable suggestions to refine this article. References B. 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