This article presents a short and concise survey on prime cordial and divisor cordial labeling of graphs. A prime cordial labeling of a graph G(V,E) is a bijective function f:V(G) → {1,2,…,|V|} such that if each edge xy is assigned the label 1 if gcd(f(x),f(y)) = 1 and 0 if gcd(f(x),f(y)) > 1, then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. Further, a divisor cordial labeling of G is a bijection g: V(G) → {1,2,…,|V|} such that an edge st is assigned the label 1 if one g(s) or g(t) divides the other and 0 otherwise, then the number of edges labeled with 0 and the number of edges labelled with 1 differ by at most 1. We call G a divisor cordial graph if it admits a divisor cordial labeling. This article stands divided into five sections. The first and fifth sections are reserved respectively for introduction and some important references. The second section deals with the prime cordial labeling of certain classes of graphs wherein some important known results have been recalled. The third section deals with the divisor cordial labeling of graphs in which a few known results of high interest have been outlined. In the fourth section we highlight certain conjectures and open problems in respect of the above mentioned labelling that still remain unsolved.

A PCL of G is a bijective map g from V to {1, 2, 3, |V|} in such a way that if an edge st is given label 1 if GCD(g(s), g(t)) = 1 & 0 otherwise, then the edges given 0 & 1 differ by at most 1 i.e; |eg (0) − eg (1)| ≤ 1. If a graph permits a PCL, then it is called a PCG. In this paper, we prove that lilly graph admits a PCL. Further, we have shown that lilly graph under some graph operations like switching of a vertex, duplication of a vertex, degree splitting graph and barycentric subdivision admits a PCL which may find its application in the development of artificial intelligence.

An assignment of intergers to the vertices of a graph Ḡ subject to certain constraints is called a vertex labeling of Ḡ. 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 Ḡ is a bijective function f from node set V of Ḡ 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 Ḡ 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 2K 1,n joining 2P n , n ≥ 2 sharing a common node. i.e., I n = 2K 1,n + 2P 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 Ḡ, 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 Ḡ.

In this paper a new variant of divisor cordial labeling (DCL) named double divisor cordial labeling (DDCL) is in-troduced. A DDCL of a graph Gω having a node set Vω is a bijection gω from Vω to {1,2,3,…, |Vω |} such that each edge yz is given the label 1 if 2gω (y)/gω (z) or 2gω (z)/gω (y) and 0 otherwise, then the modulus of difference of edges labeled 0 and those labeled 1 do not exceed 1 i.e; |eg ω (0) — eg ω (1)| ≤ 1. If a graph permits a DDCL, then it is known as double divisor cordial graph (DDCG). In this paper we derive certain general results concerning DDCL and establish the same for some well known graphs.