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 u v . The NSD total chromatic number of G , denoted by χ Σ t ( 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 χ Σ t ( G ) ≤ Δ ( G ) + 3 , where Δ ( G ) denotes the maximum degree of G . The neighbor sum distinguishing total choice number of G , denoted by ch Σ t ( 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, χ Σ t ( G ) ≤ ch Σ t ( G ) . In this paper, we prove that ch Σ t ( 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.