Neighbor sum distinguishing total coloring of sparse IC-planar graphs

Abstract Two distinct crossings are independent if the end-vertices of the crossed edge 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 ∑ 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- k -coloring of G such that for each edge u v ∈ 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 χ Σ ′ ′ ( G ) . In this paper, it is proved that χ Σ ′ ′ ( G ) ≤ max { Δ ( G ) + 3 , 11 } if G is a triangle-free IC-planar graph, and χ Σ ′ ′ ( G ) ≤ max { Δ ( G ) + 3 , 15 } if G is an IC-planar graph without adjacent triangles, where Δ ( G ) is the maximum degree of G .