Neighbor Sum Distinguishing Total Coloring Of Triangle Free Planar Graphs

Let φ:V(G)∪ E(G)→ {1,2,..,k} be a proper k-total coloring of graph G.A k-total coloring of G is proper,if any two adjacent or incident elements of G have different colors.Let 0 be a proper k-total coloring of graph G and f(v)=∑e(?)φ(e)+φ(v).We call φ is neighbor sum distinguishing k-total coloring if f(u)≠ f(v)for each edge uv ∈ E(G),and the smallest value of k is called the neighbor sum distinguishing total chromatic number,denoted by χ∑"(G).We mainly discuss neighbor sum distinguishing total coloring of triangle free planar graphs and triangle free planar graphs with restriction of maximum de-gree vertices in this paper.By using Euler’s formula,discharging methods and Combinatorial Nullstellsatz,we have the following results.Conclusion 1 Let G be a triangle free planar graph with △(G)≥ 8,thenχ∑"(G)≤△(G)+ 2.Conclusion 2 Let G be a triangle free planar graph without adjacent maxi-mum degree vertices,if △(G)>9,then χ∑"(G)= △(G)+ 1.Corollary 1 Let G be a triangle free planar graph with △(G)≥>9,if G is a graph without adjacent maximum degree vertices,then χ∑"(G)= △(G)+ 1,otherwise χ∑"(G)= △(G)+ 2.