Neighbor Sum Distinguishing Total Coloring Of Some Graphs

Let G=(V(G),E(G))be a finite,undirected and simple graph.A proper k-total coloring of a graph G is a mapping φ:V(G)U E(G)→{1,2,...,k},such that no two adjacent or incident elements of G receive the same color.For any v ∈ V(G),set f(v)denote the sum of the color of the vertex v and the colors of the edges incident with v.that is f(v)=∑uv∈E(G)(uv)+φ(v).The coloringφ is k-neighbor sum distinguishing total coloring if f(u)≠f(v)for each edge uv∈E(G).The smallest number k in such a coloring of G is the neighbor sum distinguishing total chromatic number,denoted by χ"∑(G).For neighbor sum distinguishing total chromatic number,Pilsniak and Wozniak conjectured that for any simple graph,χ"Σ(G)≤△(G)+3.In this paper,by using the famous Combinatorial Nullstellensatz and discharging method,we studied the neighbor sum distinguishing total coloring of planar graphs without 5-cycles,planar graphs without 4-cycles and 2-degenerate graphs.The following theorems are the main results of this paper.1.Let G be a planar graph without 5-cycles,then χ"∑(G)<max {△(G)+3,10}.2.Let G be a planar graph without 4-cycles and △(G)≥9.If G has no adjacent △(G)-vertices,then χ"∑(G)=△(G)+1,otherwise χ"∑(G)=△(G)+2.3.Let G be a 2-degenerate graph and △(G)≥ 6.If G has no adjacent△(G)-vertices,then χ"∑(G)=△(G)+1,otherwise χ"∑(G)=△(G)+2.