F-GEOMETRIC MEAN LABELING OF SOME ... - Kjm.pmf.kg.ac.rs

174 A. DURAI BASKAR, S. AROCKIARAJ AND B. RAJENDRANCase (i) ⌊√ 6n ⌋ is a multiple of 3.We define f : V[C n ⊙S 2 ] → {1,2,3,...,3n+1} as follows:⎧ ⌊ √6n ⌋3i−1 1 ≤ i ≤3⎪⎨ ⌊ √6n ⌋f(u i ) = 3i+1 i = +13⌊ √6n ⌋ ⎪⎩3i +2 ≤ i ≤ n,3⎧ ⌊ √6n ⌋⎪⎨ 3i−2 1 ≤ i ≤f(v (i)31 ) = ⌊ √6n ⌋⎪⎩ 3i−1 +1 ≤ i ≤ n3andf(v (i)2 ) = ⎧⎨⎩3i3i+1⌊ √6n ⌋1 ≤ i ≤ +13⌊ √6n ⌋+2 ≤ i ≤ n.3The induced edge labeling is as follows⎧⎨ 3if ∗ (u i u i+1 ) =⎩ 3i+1⌊√ ⌋f ∗ (u n u 1 ) = 6n ,⎧⎨ 3i−2f ∗ (u i v (i)1 ) =⎩ 3i−1⌊ √6n⌋1 ≤ i ≤ −13⌊ √6n ⌋≤ i ≤ n−1,3⌊ √6n⌋1 ≤ i ≤3⌊ √6n ⌋+1 ≤ i ≤ n3and⎧⎨f ∗ (u i v (i)2 ) =⎩3i−13i⌊ √6n ⌋1 ≤ i ≤3⌊ √6n ⌋+1 ≤ i ≤ n.3Case (ii) ⌊√ 6n ⌋ is not a multiple of 3.We define f : V[C n ⊙S 2 ] → {1,2,3,...,3n+1} as follows:⎧ ⌊ √6n ⌋⎨ 3i−1 1 ≤ i ≤3f(u i ) = ⌊ √6n ⌋⎩ 3i +1 ≤ i ≤ n,3⎧ ⌊ √6n+1 ⌋⎨ 3i−2 1 ≤ i ≤f(v (i)31 ) = ⌊ √6n+1 ⌋⎩ 3i−1 +1 ≤ i ≤ n3