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 u v ∈ 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 ) ( χ a ′ ( G ) ). It was conjectured that if G is a connected graph and G ∉ { K 2 , C 5 } , then χ ∑ ′ ( G ) ≤ Δ ( G ) + 2 and χ a ′ ( G ) ≤ Δ ( G ) + 2 . For a given graph G , let ( L e ) 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 L e for each e ∈ E is called the list neighbor sum (set) distinguishing index of G , and denoted by ch ∑ ′ ( G ) ( ch a ′ ( 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 ch a ′ ( G ) ≤ Δ ( G ) + 3 . This improves a result by Przybylo and Wong (Przybylo and Wong, 2015), which states that if G is a planar graph without isolated edges, then ch ∑ ′ ( G ) ≤ Δ ( G ) + 13 (so ch a ′ ( G ) ≤ Δ ( G ) + 13 also holds). Our approach is based on the Combinatorial Nullstellensatz and the discharging method.

中文翻译：

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改进了邻居总和（集）的界限，以区分平面图的可选项性

摘要 令 G = ( V , E ) 是一个简单的图， ϕ : E ( G ) → { 1 , 2 , … , k } 是 G 的适当 k 边着色。我们说 ϕ 是邻居和（集），如果对于每条边 uv ∈ E ( G ) ，在与 u 入射的边上所取的颜色总和（集）不同于边上所取颜色的总和（集）与 v 的事件。使 G 具有区分 k 边缘着色的邻居和（集）的最小 k 称为 G 的邻居和（集）区分索引，用 χ ∑ ′ ( G ) ( χ a ′ ( G ) ) 表示。推测如果G是连通图且G∉{K2,C5}，则χ∑′(G)≤Δ(G)+2且χa′(G)≤Δ(G)+2。对于给定的图 G ，让 (L e ) e ∈ E 是一组实数列表，每个列表的大小为 k 。最小的 k 使得对于此类列表的任何指定集合，存在使用来自 Le e 的颜色对每个 e ∈ E 区分边缘着色的邻居和（集）称为 G 的列表邻居和（集）区分索引，并表示为ch ∑ ′ ( G ) ( ch a ′ ( G ) ) 。在本文中，我们证明如果 G 是一个 Δ ( G ) ≥ 22 且没有孤立边的平面图，则 ch ∑ ′ ( G ) ≤ Δ ( G ) + 6 且 ch a ′ ( G ) ≤ Δ ( G)+3。这改进了 Przybylo 和 Wong（Przybylo 和 Wong，2015 年）的结果，该结果指出，如果 G 是没有孤立边的平面图，则 ch ∑ ′ ( G ) ≤ Δ ( G ) + 13 (so ch a ′ ( G ) ≤ Δ ( G ) + 13 也成立）。我们的方法基于组合 Nullstellensatz 和放电方法。