Neighbor Sum Distinguishing Total Choice Number of IC-Planar Graphs Without 4-Cycles

A neighbor sum distinguishing (NSD) total coloring of a graph G is a mapping ϕ:T(G)=V(G)∪E(G)→{1,2,⋯,k} such that any two adjacent or incident elements in T(G) receive different colors, and the sum of the colors of all incident edges of u and the color of u is different from the sum of the colors of all incident edges of v and the color of v for each edge uv. The NSD total chromatic number of G, denoted by χΣt(G), is the smallest integer k such that G has an NSD total coloring. For any graph G, there is a conjecture that the NSD total chromatic number χΣt(G)≤Δ(G)+3, where Δ(G) denotes the maximum degree of G. The neighbor sum distinguishing total choice number of G, denoted by chΣt(G), is the smallest integer k such that, after assigning each z∈T(G) a set L(z) of k real numbers, G has an NSD total coloring ϕ satisfying ϕ(z)∈L(z) for each z∈T(G). Obviously, χΣt(G)≤chΣt(G). In this paper, we prove that chΣt(G)≤Δ(G)+3 for any IC-planar graph G without 4-cycles and Δ(G)≥7 by applying the Combinatorial Nullstellensatz, which improves upon the previous results.