Odd- Even Congruence Labeling of Certain Classes of Graphs

Abstract

In graph theory, graph labeling involves the systematic assignment of positive integers to the structural components of a graph, specifically its vertices and edges, under predefined constraints. When modular arithmetic is utilized to assign odd values to vertices and even values to edges, the scheme is identified as odd even congruence labeling. Formally, a graph G=(V,E)admits an odd-even congruence labeling if there exist injective mappings h: V(G) → ℤ+and k: E(G) → ℤ +such that for every edge e=wz ∈ E(G), the relation h(w) ≡h(z) (mod k(e))holds. This study investigates the implementation of this technique across several unique architectures. Specifically, we formally prove that the ice cream graph, umbrella graph, double fan graph, rectangular book graph, pentagonal book graph, path graph, triangular ladder graph, quadrilateral snack graph, jewel graph, Vanessa graph, helm graph, closed helm graph, jellyfish graph, andsunlet graphall admits an odd even congruence labeling.