2-Odd Labeling of Graphs Using Certain Number Theoretic Concepts and Graph Operations

Abstract

Graph theory plays a significant role in a variety of real-world systems. Graph concepts such as labeling and coloring are used to depict a variety of processes and relationships in material, social, biological, physical, and information systems. Specifically, graph labeling is used in communication network addressing, fault-tolerant system design, automatic channel allocation, etc. 2-odd labeling assigns distinct integers to the nodes of <img src=image/13427595_01.png> in such a manner, that the positive difference of adjacent nodes is either 2 or an odd integer, <img src=image/13427595_02.png>, <img src=image/13427595_03.png>. So, <img src=image/13427595_04.png> is a 2-odd graph if and only if it permits 2-odd labeling. Studying certain important modifications through various graph operations on a given graph is interesting and challenging. These operations mainly modify the underlying graph's structure, so understanding the complex operations that can be done over a graph or a set of graphs is inevitable. The motivation behind the development of this article is to apply the concept of 2-odd labeling on graphs generated by using various graph operations. Further, certain results on 2-odd labeling are also derived using some well-known number theoretic concepts such as the Twin prime conjecture and Goldbach's conjecture, besides recalling a few interesting applications of graph labeling and graph coloring.