Neighbor sum distinguishing total choosability of 1-planar graphs with maximum degree at least 24

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 .