Neighbor Sum Distinguishing Total Choosability Of 1-Planar Graphs With Maximum Degree At Least 24

For a simple graph G, a neighbor sum distinguishing total k-coloring of G is a mapping phi: V(G)boolean OR E(G) -> {1, 2, . . . , k} such that no two adjacent or incident elements in V(G)boolean OR E(G) receive the same color and w phi(u) not equal w phi(v) for each edge uv is an element of E(G), where w phi(v) (or w phi(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 is an element of V(G)boolean OR E(G), let L(x) be a list of integer numbers. If, whenever we give a list assignment L = {L(x)parallel to L(x)vertical bar= k, x is an element of V(G)boolean OR E(G)}, there exists a neighbor sum distinguishing total k-coloring phi such that phi(x) is an element of L(x) for each element x is an element of V(G)boolean OR E(G), then we say that phi 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 ch(Sigma)''. 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 ch(Sigma)''(G) if G is a 1-planar graph. We prove that ch(Sigma)''(G) <= Delta + 3 for every 1-planar graph G with maximum degree Delta >= 24. (C) 2020 Elsevier B.V. All rights reserved.