Neighbor Sum Distinguishing Total Coloring Of Planar Graphs

A proper k-total coloring of a graph G is a map φ: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)φ(uv)+φ(v).φ 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 byχ∑"(G).For neighbor sum distinguishing total chromatic number,Pilsniak and Wozniak conjectured that for any simple graph with maximum degree △(G),χ∑"(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,χ∑"(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 χ∑"(G)=△(G)+ 1 if G contains no adjacent △-vertices,otherwise χ∑"(G)= △(G)+ 2.Conclusion 3 Let G be a planar graph with maximum degree △(G),thenχ∑"(G)<max{△(G)+ 2,14}.Conclusion 4 Let G be a planar graph without 4-cycles,then χ∑"(G)≤max{△(G)+ 2,10}.