On Recent Advances in Divisor Cordial Labeling of Graphs

Abstract

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.