Edge odd graceful of alternate snake graphs

Abstract

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, …,2q − 1} so that induced mapping f ∶ V(G) → {0, 1,2, …,2q − 1} given by f (x) = ∑xy∈E(G) f(xy) (mod 2q) is injective. A graph which admits an edge odd graceful labeling is called edge odd graceful. An alternate triangular snake graph A(C3m) is a graph obtained from a path u1u2u3 … u2m by joining every u2i-1 and u2i to a new vertex vi, 1 ≤ i ≤ m. An alternate quadrilateral snake graph A(C4m) is a graph obtained from vertices u1, u2, u3, …, u2m by joining every u2i-1 and u2i to two vertices vi and wi, 1 ≤ i ≤ m, and joining every u2 to u2i-1 with 1 ≤ i ≤ m − 1. In this paper, we show that alternate triangular snake and alternate quadrilateral snake graphs are edge odd graceful.