On Recent Advances in Divisor Cordial Labeling of Graphs

An assignment of intergers to the vertices of a graph ¯ G subject to certain constraints is called a vertex labeling of ¯ G . Different types of graph labeling techniques are used in the ﬁeld of coding theory, cryptography, radar, missile guidance, x -ray crystallography etc. A DCL of ¯ G is a bijective function ¯ f from node set ¯ V of ¯ G to { 1 , 2 , 3 , ..., | ¯ V |} such that for each edge rs , we allot 1 if ¯ f ( r ) divides ¯ f ( s ) or ¯ f ( s ) divides ¯ f ( r ) & 0 otherwise, then the absolute difference between the number of edges having 1 & the number of edges having 0 do not exceed 1 , i.e., | e ¯ f (0) − e ¯ f (1) | ≤ 1 . If ¯ G permits a DCL, then it is called a DCG. A complete graph K n , is a graph on n nodes in which any 2 nodes are adjacent and lilly graph I n is formed by 2 K 1 ,n joining 2 P n , n ≥ 2 sharing a common node. i.e., I n = 2 K 1 ,n + 2 P n , where K 1 ,n is a complete bipartite graph & P n is a path on n nodes. In this paper, we propose an interesting conjecture concerning DCL for a given ¯ G , besides, discussing certain general results concerning DCL of complete graph K n − related graphs. We also prove that I n admits a DCL for all n ≥ 2 . Further, we establish the DCL of some I n − related graphs in the context of some graph operations such as duplication of a node by an edge, node by a node, extension of a node by a node, switching of a node, degree splitting graph, & barycentric subdivision of the given ¯ G .


Introduction
ByḠ, we denote a simple, finite, & undirected graph with node setV & edge setĒ. An allocation of labels to nodes or edges or sometimes both, under some constraints is called as graph labeling. Graph labeling is a close association of graph theory & number theory. Being interdisciplinary, graph labeling is attracting the attention of numerous researchers and software developers. For number theory and graph theory related terms, we refer to [1] and [4], respectively. For further study on various graph labeling problems, see [3]. We use DCL and DCG to denote divisor cordial labeling and divisor cordial graph, respectively.
Cahit [2] introduced the idea of cordial labeling. Sundaram et al. [9] coined the notion of prime cordial labeling. The concept of DCL was given by Vartharajan et al. [10]. Vartharajan et al. [11] proved some general results especially the DCL of full binary tree. Definition 1. [10] A DCL ofḠ havingV is a bijectionf fromV to {1, 2, 3, ..., |V |} such that each edge rs is alloted IfḠ admits a DCL, then it is said to be a DCG.

Main Results
This section is devoted to derive some general results on DCG. Also, DCL of lilly graph in the context of different graph operations has been explored.

DCL of K n Related Graphs
Let N (u) and N [u] represent the open and closed neighbourhood of u, respectively. In this section, we deal with K n related graphs in establishing DCL.
where eachS i is a set of nodes having at least two nodes of same degree &T = V − S i . DS(Ḡ) is the degree splitting graph ofḠ, which is created fromḠ by adding nodes w 1 ,w 2 , ...,w t & connectingw i to each node ofS i ; 1 ≤ i ≤ t.
[10] K n does not admit a DCL for n ≥ 7.
Theorem 2. DS(K n ) does not permit a DCL for n ≥ 6.
Proof. The proof follows clearly from Theorem 1 and Lemma 1.
Lemma 2. Extension of any arbitrary node of K n yields K n+1 .
The proof follows from the fact that the newly added node is joined with all the nodes of K n including the node itself as every pair of nodes in K n are adjacent, which eventually gives rise to K n+1 . Theorem 3. The graph G obtained by performing extension of any arbitrary node in K n does not admit a DCL for n ≥ 6.
Proof. It follows from Theorem 1 and Lemma 2. Lemma 3. The graph formed by switching any arbitrary node in K n admits a DCL for n ≤ 8.
Proof. Switching of any arbitary node in K n results in a disconnected graph whose components are K n−1 and K 1 . The result clearly follows for switching of node in K n for n = 3, 4, 6, 7 (see Figure 1). Now we discuss DCL of switching of a node in K 5 and K 8 . Case(i). When n = 5. Label the isolated node with 4 and assign the remaining labels to the nodes of K 4 . Clearly, e(0) = e(1) = 3. Case(ii). When n = 8. Label the isolated node with 7 and assign the remaining labels to the nodes of K 7 . Here e(0) = 10 and e(1) = 11. Theorem 4. Switching of an arbitrary node in K n for n ≥ 9 does not admit a DCL.
Proof. Switching of an arbitrary node in K n ; n ≥ 9 results in a disconnected graph G whose components are K n−1 and K 1 . We take n = 9 for the sake of discussion. We obtain a disconnected G obtained by switching of a node in K 9 whose Figure 1. DCL of switching of a node in K 4 , K 6 and K 7 componenets are K 8 and K 1 . We prove by a method of contradiction. We assume that G admits a DCL. Without loss of generality, label the isolated node with the largest prime p where p ≤ 9 (i.e., 7) in order to get more edges having label 1, and assign the remaining labels to nodes of K 8 in any order. Here e(0) = 15, e(1) = 13, and therefore |e(0) − e(1)| > 1, a contradiction. The other possibilities of assigning different labels to the isolated node can be dealt in the similar lines. The similar argument holds good for n ≥ 10. Hence the theorem.
Considering the fact that characterization of DCGs is challenging in general, we propose the following conjecture.
Conjecture 1. For a given finite graphḠ, establishing a DCL ofḠ is NP-hard.
Remark 1. We believe that the conjecture is true as there are no algorithm available in the literature and devising a particular pattern of DCG is also the hardest.

DCL of Lilly Related Graphs
Here, we explore the DCL of lilly graph and its related graphs in the context of various graph operations.
Theorem 10. The duplication of an arbitrary node of degree 1 or 2 in I n permits a DCL.
Proof. LetḠ be formed by duplicating any arbitrary node x k of I n by the the newly inserted node s. Then the cardinality of node set ofḠ is 4n. Now arise two cases. Case(i). Duplication of a node of degree 1. In this case, the cardinality of edge set ofḠ is 4n − 1. Define a function (bijective) T :V (Ḡ) → {1, 2, ..., 4n} by fixing T (x 3n ) = 2, T (x 1 ) = 1 and T (s) be the largest prime p ≤ 4n. Assign the unutilized even labels to x i ; 2 ≤ i ≤ 2n and odd labels simultaneously to x j ; 2n + 1 ≤ j ≤ 4n − 1, j = 3n. Case(ii). Duplication of a node of degree 2. In this case, the cardinality of edge set ofḠ is 4n. Labeling is done by using the pattern of case(i) (see Figure 5). In both the cases, we observe that the difference of edges having labels 1 and 0 is not more than 1 which verifies thatḠ is a DCG.  T (x 2n+1 ) = 6. Assign the unutilized labels to the remaining nodes in any order. It follows thatḠ is a DCG (see Figure  6). Proof. LetḠ represent the barycentric subdivision of I n .
Conclusion: This paper has dealt with certain interesting general results on DCL for K n related graphs besides, formulating an impressive conjecture on DCL. Further, we have proved that lilly graph admits a DCL and discussed DCL for some lilly related graphs under various graph operations.