Neighbor Sum Distinguishing Total Coloring Of Planar Graphs

A proper k-total coloring of a graph G is a map ?:V(G)UE(G)?{1,2,…,k},such that any two adjacent or incident elements of G have different colors.Let f(v)=?uv?E(G)?(uv)+?(v).? is a k-neighbor sum distinguishing total coloring of graph 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 by??"(G).For neighbor sum distinguishing total chromatic number,Pilsniak and Wozniak conjectured that for any simple graph with maximum degree ?(G),??"(G)<?(G)+ 3.The aim of this paper is to consider neighbor sum distinguishing total coloring of planar graphs,and by using the famous Combinatorial Nullstellensatz and discharging method we get the following conclusions.Conclusion 1 Let G be a Halin graph,then and if ?(G)>6,??"(G)= ?(G)+ 2 if and only if G contains two adjacent?-vertices.Conclusion 2 Let G be a K4-minor free graph with ?(G)? 5,then ??"(G)=?(G)+ 1 if G contains no adjacent ?-vertices,otherwise ??"(G)= ?(G)+ 2.Conclusion 3 Let G be a planar graph with maximum degree ?(G),then??"(G)<max{?(G)+ 2,14}.Conclusion 4 Let G be a planar graph without 4-cycles,then ??"(G)?max{?(G)+ 2,10}.