Neighbor sum distinguishing total choice number of NIC-planar graphs with restricted conditions

A NIC-planar graph is a graph that has a drawing in the plane such that each edge is crossed at most once and any two pairs of crossing edges share at most one common vertex. Let E-c(u) denote the set of edges incident with a vertex u. A neighbor sum distinguishing (NSD) total coloring phi of G is a proper total coloring of G such that Sigma(z is an element of EG <mu >phi(z)) not equal Sigma(z is an element of EGf < v >boolean OR phi(z))or each edge uv is an element of E(G). Pilsniak and Wozniak conjectured that any graph with maximum degree Delta admits an NSD total (Delta + 3) -coloring. In this paper, we prove that the list version of the conjecture holds for any triangle-free NIC-planar graph with Delta >= 8 and with each vertex incident with at most two crossing edges by applying the Combinatorial Nullstellensatz.