Abstract For a simple graph G , a neighbor sum distinguishing total k -coloring of G is a mapping ϕ : V ( G ) ∪ E ( G ) → { 1 , 2 , … , k } such that no two adjacent or incident elements in V ( G ) ∪ E ( G ) receive the same color and w ϕ ( u ) ≠ w ϕ ( v ) for each edge u v ∈ E ( G ) , where w ϕ ( v ) (or w ϕ ( u ) ) denotes the sum of the color of v (or u ) and the colors of all edges incident with v (or u ). For each element x ∈ V ( G ) ∪ E ( G ) , let L ( x ) be a list of integer numbers. If, whenever we give a list assignment L = { L ( x ) | | L ( x ) | = k , x ∈ V ( G ) ∪ E ( G ) } , there exists a neighbor sum distinguishing total k -coloring ϕ such that ϕ ( x ) ∈ L ( x ) for each element x ∈ V ( G ) ∪ E ( G ) , then we say that ϕ is a list neighbor sum distinguishing total k -coloring. The smallest k for which such a coloring exists is called the neighbor sum distinguishing total choosability of G , denoted by c h ∑ ′ ′ ( G ) . A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. There is almost no result yet about c h ∑ ′ ′ ( G ) if G is a 1-planar graph. We prove that c h ∑ ′ ′ ( G ) ≤ Δ + 3 for every 1-planar graph G with maximum degree Δ ≥ 24 .

中文翻译：

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区分最大度数至少为 24 的 1 平面图的总选择性的邻居和

摘要 对于简单图 G ，区分 G 的总 k 着色的邻居和是一个映射 ϕ : V ( G ) ∪ E ( G ) → { 1 , 2 , … , k } 使得在V ( G ) ∪ E ( G ) 接收相同的颜色并且 w ϕ ( u ) ≠ w ϕ ( v ) 对于每条边 uv ∈ E ( G ) ，其中 w ϕ ( v ) (或 w ϕ ( u ) ) 表示v（或 u ）的颜色与所有与 v（或 u ）相关的边缘颜色的总和。对于每个元素 x ∈ V ( G ) ∪ E ( G ) ，让 L ( x ) 是一个整数列表。如果，每当我们给一个列表赋值 L = { L ( x ) | | L ( x ) | = k , x ∈ V ( G ) ∪ E ( G ) } ，对于每个元素 x ∈ V ( G ) ∪ E ( G ) ，那么我们说 ϕ 是区分总 k 着色的列表邻居和。存在这种着色的最小 k 称为区分 G 的总选择性的邻居和，用 ch ∑ ′ ′ ( G ) 表示。如果一个图可以在平面上绘制，使得每条边最多与另一条边交叉，则该图是 1 平面的。如果 G 是 1-平面图，则几乎没有关于 ch ∑ ′ ′ ( G ) 的结果。我们证明 ch ∑ ′ ′ ( G ) ≤ Δ + 3 对于每个最大度数 Δ ≥ 24 的 1-平面图 G。