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

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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 EG (u) denote the set of edges incident with a vertex u. A neighbor sum distinguishing (NSD) total coloring ϕ of G is a proper total coloring of G such that for each edge uυ ∈E(G) . Pilśniak and Woźniak conjectured that any graph with maximum degree Δ admits an NSD total (Δ+ 3) -coloring. In this paper, we prove that the list version of the conjecture holds for any triangle-free NIC-planar graph with Δ ≥ 8 and with each vertex incident with at most two crossing edges by applying the Combinatorial Nullstellensatz.