The Edge Odd Graceful Labeling of Water Wheel Graphs

A graph, G=(V,E), is edge odd graceful if it possesses edge odd graceful labeling. This labeling is defined as a bijection g:E(G)→{1,3,…,2m−1}, from which an injective transformation is derived, g*:V(G)→{1,2,3,…,2m−1}, from the rule that the image of u∈V(G) under g* is ∑uv∈E(G)g(uv)mod(2m). The main objective of this manuscript is to introduce new classes of planar graphs, namely water wheel graphs, WWn; triangulated water wheel graphs, TWn; closed water wheel graphs, CWn; and closed triangulated water wheel graphs, CTn. Furthermore, we specify conditions for these graphs to allow for edge odd graceful labelings.