ABSTRACT
Let 𝐺 = (𝑉(𝐺),𝐸(𝐺)) be a graph and Let ℎ∶ 𝑉 (𝐺) → 𝑆3 be a function. For each edge 𝑥𝑦 assign the label r where r is the remainder when 𝑜(ℎ(𝑥)) is divided by 𝑜(ℎ(𝑦)) or 𝑜(ℎ(𝑦)) is divided by 𝑜(ℎ(𝑥)) according as 𝑜(ℎ(𝑥)) ≥ 𝑜(ℎ(𝑦)) or 𝑜(ℎ(𝑦)) ≥ 𝑜(ℎ(𝑥)). The function ℎ is called a group 𝑆3 cordial remainder labeling of 𝐺 if |𝑣ℎ(𝑖) − 𝑣ℎ(𝑗)| ≤ 1 and |𝑒ℎ(1) − 𝑒ℎ(0)| ≤ 1, where 𝑣ℎ(𝑖) denotes the number of vertices labeled with j and 𝑒ℎ(𝑖) denotes the number of edges labeled with 𝑖 (𝑖 = 0,1). A graph 𝐺 which admits a group 𝑆3 cordial remainder labeling is called a group 𝑆3 cordial remainder graph. In this paper, we prove that duplication of each vertex by a new vertex in star related graph admits a group 𝑆3 cordial remainder labeling.
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