Neighbor sum distinguishing total choosability of IC-planar graphs
Yuanyuan Duan, Lianying Miao, Wenyao Song · Discussiones Mathematicae Graph Theory · 2018
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-kcoloring 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 c (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-kcoloring of G such that for each edge uv E(G), c (u) = c (v). The least number k needed for such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by (G). In this paper, it is proved that if G is an IC-planar graph with maximum degree (G), then ch (G) max{(G) + 3, 17}, where ch (G) is the neighbor sum distinguishing total choosability of G.