Neighbor Sum Distinguishing Total Coloring Of Planar Graphs With Cycle Conditions

Let graph G be undirected,finite and simple.If any two incident or adjacent vertices and edges of G receive different colors denoted by natural number,then such a coloring is a proper total coloring of graph G.Furthermore,for any adjacent vertices u and v,if the sum of colors of vertex u and edges incident with u is different from the sum of colors of vertex v and edges incident with v.then such a coloring is a neighbor sum distinguishing total coloring of graph G.The smallest value of the number of colors needed in such a coloring of G is called the neighbor sum distinguishing total chromatic number χ"∑(G).In 2015,for neighbor sum distinguishing total chromatic number of graph G,Pilsniak and Wozniak conjectured that for any simple graph with maxi-mum degree Δ(G),χ"∑(G)≤Δ(G)+3.By using Combinatorial Nullstellen-satz and discharging method,we study the neighbor sum distinguishing to-tal coloring of planar graphs without 5-cycles and without 6-cycles,and get the following conclusions:1.Let G be a planar graph without 5-cycles,thenχ"∑(G)≤max{Δ(G)+2,10};2.Let G be a planar graph without 5-cycles and maximum degree at least is 9.If G has no adjacent Δ(G)-vertices,thenχ"∑(G)=Δ(G)+1,otherwise χ"∑(G)=Δ(G)+2;3.Let G be a planar graph without 6-cycles,then χ"∑(G)≤max{Δ(G)+3,10}.Conclusions 1 and Con-clusions 2 mean Pilsniak and Wozniak’s conjecture is true for planar graph G without 5-cycles.At the same time,the bound of neighbor sum distinguishing total chromatic number is sharp.Conclusions 3 means Pilsniak and Wozniak’s conjecture is true for planar graph G without 6-cycles.