On generalized neighbor sum distinguishing index of planar graphs

Abstract

For a proper [Formula: see text]-edge coloring [Formula: see text] of a graph [Formula: see text], let [Formula: see text] denote the sum of the colors taken on the edges incident to the vertex [Formula: see text]. Given a positive integer [Formula: see text], the [Formula: see text]-neighbor sum distinguishing [Formula: see text]-edge coloring of G is [Formula: see text] such that for each edge [Formula: see text], [Formula: see text]. We denote the smallest integer [Formula: see text] in such coloring of [Formula: see text] by [Formula: see text]. For [Formula: see text], Wang et al. proved that [Formula: see text]. In this paper, we show that if G is a planar graph without isolated edges, then [Formula: see text], where [Formula: see text].