Improved bounds for neighbor sum (set) distinguishing choosability of planar graphs

Abstract

Let G=(V,E) be a simple graph and ϕ:E(G)→{1,2,…,k} be a proper k-edge coloring of G. We say that ϕ is neighbor sum (set) distinguishing if for each edge uv∈E(G), the sum (set) of colors taken on the edges incident with u is different from the sum (set) of colors taken on the edges incident with v. The smallest k such that G has a neighbor sum (set) distinguishing k-edge coloring is called the neighbor sum (set) distinguishing index of G and denoted by χ∑′(G) (χa′(G)). It was conjectured that if G is a connected graph and G∉{K2,C5}, then χ∑′(G)≤Δ(G)+2 and χa′(G)≤Δ(G)+2. For a given graph G, let (Le)e∈E be a set of lists of real numbers and each list has size k. The smallest k such that for any specified collection of such lists there exists a neighbor sum (set) distinguishing edge coloring using colors from Le for each e∈E is called the list neighbor sum (set) distinguishing index of G, and denoted by ch∑′(G)(cha′(G)). In this paper, we prove that if G is a planar graph with Δ(G)≥22 and with no isolated edges, then ch∑′(G)≤Δ(G)+6 and cha′(G)≤Δ(G)+3. This improves a result by Przybyło and Wong (Przybyło and Wong, 2015), which states that if G is a planar graph without isolated edges, then ch∑′(G)≤Δ(G)+13 (so cha′(G)≤Δ(G)+13 also holds). Our approach is based on the Combinatorial Nullstellensatz and the discharging method.