Edge odd graceful of alternate snake graphs

Let G be a graph with vertex set V ( G ), edge set E ( G ), and the number of edges q . An edge odd graceful labeling of G is a bijection f : E ( G ) → {1,3,5, …,2 q − 1} so that induced mapping f + : V ( G ) → {0,1,2, …,2 q − 1} given by f + ( x ) = ∑ xy ∈ E ( G ) f ( xy ) ( mod 2 q ) is injective. A graph which admits an edge odd graceful labeling is called edge odd graceful. An alternate triangular snake graph A ( C 3 m ) is a graph obtained from a path u 1 u 2 u 3 … u 2 m by joining every u 2 i −1 and u 2 i to a new vertex υ i , 1 ≤ i ≤ m . An alternate quadrilateral snake graph A ( C 4 m ) is a graph obtained from vertices u 1 , u 2 , u 3 ,…, u 2 m by joining every u 2 i −1 and u 2 i to two vertices υ i and w i ,1 ≤ i ≤ m , and joining every u 2 i to u 2 i +1 with 1 ≤ i ≤ m − 1. In this paper, we show that alternate triangular snake and alternate quadrilateral snake graphs are edge odd graceful.