Arithmetic Sequential Graceful Labeling For Arrow Graphs

Abstract

Let G be a simple, finite, connected, undirected, non-trivial graph with 𝑝 vertices and 𝑞 edges. 𝑉(𝐺) be the vertex set and 𝐸(𝐺) be the edge set of 𝐺. Let 𝑓: 𝑉(𝐺) → {𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, 𝑎 + 3𝑑, … , 𝑎 + 2𝑞𝑑} where a ≥ 0 and 𝑑 ≥ 1 is an injective function. If for each edge 𝑢𝑣 ∈ 𝐸(𝐺) , 𝑓 ∗ : 𝐸(𝐺) → {𝑑, 2𝑑, 3𝑑, 4𝑑, … , 𝑞𝑑} defined by 𝑓 ∗ (𝑢𝑣) = |𝑓(𝑢)− 𝑓(𝑣)| is a bijective function then the function 𝑓 is called arithmetic sequential graceful labeling. The graph with arithmetic sequential graceful labeling is called arithmetic sequential graceful graph. In this paper, we prove that one side arrow graphs 𝐴𝑅𝜂 2 , 𝐴𝑅𝜂 3 , 𝐴𝑅𝜂 5 and double-sided arrow graphs 𝐷(𝐴𝑅𝜂 2 ),𝐷(𝐴𝑅𝜂 3 ) are arithmetic sequential graceful graph