On generalized neighbor sum distinguishing index of planar graphs.

For a proper k -edge coloring ϕ : E (G) → { 1 , 2 , ... , k } of a graph G , let w (v) denote the sum of the colors taken on the edges incident to the vertex v. Given a positive integer p , the Σ p -neighbor sum distinguishing k -edge coloring of G is ϕ such that for each edge u v ∈ E (G) , | w (v) − w (u) | ≥ p. We denote the smallest integer k in such coloring of G by χ Σ p ′ (G). For p = 1 , Wang et al. proved that χ Σ 1 ′ (G) ≤ max { Δ (G) + 1 0 , 2 5 }. In this paper, we show that if G is a planar graph without isolated edges, then χ Σ p ′ (G) ≤ max { Δ (G) + (1 6 p − 6) , f (p) } , where f (p) = max { 2 2 p + 3 , 8 p 2 + 2 6 p + 1 + (2 p + 1) 1 6 p 2 + 9 6 p − 1 5 4 }. [ABSTRACT FROM AUTHOR]