On Recent Advances in Divisor Cordial Labeling of Graphs

Vishally Sharma,A. Parthiban

doi:10.13189/ms.2022.100111

Abstract

An assignment of intergers to the vertices of a graph <img src=image/13424638_01.gif> subject to certain constraints is called a vertex labeling of <img src=image/13424638_01.gif>. Different types of graph labeling techniques are used in the field of coding theory, cryptography, radar, missile guidance, <img src=image/13424638_02.gif>-ray crystallography etc. A DCL of <img src=image/13424638_01.gif> is a bijective function <img src=image/13424638_03.gif> from node set <img src=image/13424638_04.gif> of <img src=image/13424638_01.gif> to <img src=image/13424638_05.gif> such that for each edge <img src=image/13424638_06.gif>, we allot 1 if <img src=image/13424638_07.gif> divides <img src=image/13424638_08.gif> or <img src=image/13424638_08.gif> divides <img src=image/13424638_07.gif> & 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., <img src=image/13424638_09.gif>. If <img src=image/13424638_01.gif> permits a DCL, then it is called a DCG. A complete graph <img src=image/13424638_10.gif>, is a graph on <img src=image/13424638_11.gif> nodes in which any 2 nodes are adjacent and lilly graph <img src=image/13424638_12.gif> is formed by <img src=image/13424638_13.gif> joining <img src=image/13424638_14.gif>, <img src=image/13424638_15.gif> sharing a common node. i.e., <img src=image/13424638_16.gif>, where <img src=image/13424638_17.gif> is a complete bipartite graph & <img src=image/13424638_18.gif> is a path on <img src=image/13424638_11.gif> nodes. In this paper, we propose an interesting conjecture concerning DCL for a given <img src=image/13424638_01.gif>, besides, discussing certain general results concerning DCL of complete graph <img src=image/13424638_10.gif>-related graphs. We also prove that <img src=image/13424638_12.gif> admits a DCL for all <img src=image/13424638_15.gif>. Further, we establish the DCL of some <img src=image/13424638_12.gif>-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 <img src=image/13424638_01.gif>.

Highlights

By G, we denote a simple, finite, & undirected graph with node set V & edge set E
We propose an interesting conjecture concerning DCL for a given G, besides, discussing certain general results concerning DCL of complete graph Kn−related graphs
[6] Let Gwith V (G) = S1 ∪ S2, ... ∪ St ∪ T, where each Si is a set of nodes having at least two nodes of same degree & T = V − Si

Summary

Introduction

By G, we denote a simple, finite, & undirected graph with node set V & edge set E. The graph G obtained by performing extension of any arbitrary node in Kn does not admit a DCL for n ≥ 6. The graph formed by switching any arbitrary node in Kn admits a DCL for n ≤ 8. The result clearly follows for switching of node in Kn for n = 3, 4, 6, 7 (see Figure 1).

Results

Conclusion