Neighbor Sum Distinguishing Total Coloring Of Two Kinds Of Planar Graphs

Let φ:V(G)∪E(G)→{1,2,...,k} be a proper k-total coloring of a graph G such that any two adjacent or incident elements of G have different colors.Let f(v)=∑uv∈E(G)φ(uv)+φ(v).The coloring 0 is a k-neighbor sum distinguishing total coloring of G if f(u)≠f(v)for each edge uv∈E(G),and the smallest value k is called the neighbor sum distinguishing total chromatic number,denoted byχ"∑(G).We mainly discuss that the neighbor sum distinguishing total colorings of pla-nar graphs with girth at least 5 and planar graphs without maximum degree vertices in this paper.By using Euler’s formula,Combinatorial Nullstellensatz and discharging method,we get the following results.1.Let G be a planar graph with g(G)≥>5,then χ"∑(G)≤max{△(G)+2,8}.2.Let G be a planar graph with g(G)≥5 and △(G)≥7,If G has no adjacent△(G)-vertices,then χ"∑(G)=△(G)+1,otherwise χ"∑(G)=△(G)+2.3.Let G be a planar graph and △(G)≥ 13,If G has no adjacent △(G)-vertices,then χ"∑(G)=△(G)+1,otherwise χ"∑(G)=△(G)+2.