Edge Odd Graceful Labeling in Some Wheel-Related Graphs

A graph’s edge labeling involves the allocation of symbols (colors or numbers) to the edges of a graph governed by specific criteria. Such labeling of a graph G with order n and size m is named edge odd graceful if there is a bijective map φ from the set of edges E ( G ) = { e 1 , … , e m } to the set { 1 , 3 , … , 2 m − 1 } in a way that the derived transformation φ ∗ from the vertex-set V ( G ) = { v 1 , … , v n } to the set { 0 , 1 , 2 , … , 2 m − 1 } given by φ ∗ ( u ) = ∑ u v ∈ E ( G ) φ ( u v ) mod ( 2 m ) is injective. Any graph is named edge odd graceful if it permits an edge odd graceful allocation (Solairaju and Chithra). The primary aim of this study is to define and explore the edge odd graceful labeling of five new families of wheel-related graphs. Consequently, necessary and sufficient conditions for these families to be edge odd graceful are provided.