Neighbor sum distinguishing total choosability of IC-planar graphs

Two distinct crossings are independent if the end-vertices of the crossed pair of edges are mutually different. If a graph G has a drawing in the plane such that every two crossings are independent, then we call G a plane graph with independent crossings or IC-planar graph for short. A proper total-k-coloring of a graph G is a mapping c : V (G) ∪ E(G) → {1, 2, . . . , k} such that any two adjacent elements in V (G) ∪ E(G) receive different colors. Let Σc(v) denote the sum of the color of a vertex v and the colors of all incident edges of v. A total-k-neighbor sum distinguishing-coloring of G is a total-k-coloring of G such that for each edge uv ∈ E(G), Σc(u) ≠ Σc(v). The least number k needed for such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by χΣ″(G) \chi _\Sigma ^{''} ( G ) . In this paper, it is proved that if G is an IC-planar graph with maximum degree Δ(G), then chΣ″(G)≤max{Δ(G)+3, 17} ch_\Sigma ^{''} ( G ) \le \max \left\{ {\Delta ( G ) + 3,\;17} \right\} , where chΣ″(G) ch_\Sigma ^{''} ( G ) is the neighbor sum distinguishing total choosability of G.