The Linear 2-arboricity Of A Kind Of Graph And Neighbor Sum Distinguishing Total Coloring Of A Kind Of Ic-planar Graph

Let G=(V(G),E(G))be a finite,undirected and simple graph with maximum degree ?(G).The linear 2-arboricity of graph G is the least integer m such that G can be partitioned into m edge disjoint linear 2-forests whose component trees are paths of length at most 2.A proper k-total coloring of a graph G is a mapping?:V(G)U E(G)? {1,2,…,k},such that any two adjacent or incident elements of G have different colors.Suppose f(v)=?uv?E(G)?(uv)+?(v).For each edge uv ? E(G),if f(u)?f(v),? is a k-neighbor sum distinguishing total coloring of G.The smallest number k in such a coloring of G is called the neighbor sum distinguishing total chromatic number,denoted by x"?(G).We mainly studied the linear 2-arboricity of graphs with mad(G)?10/3 and neighbor sum distinguishing total coloring of triangle free IC-planar graphs in this paper.The linear 2-arboricity of graphs with mad(G)?10/3 are studied in this paper.We get the following result.Let G be a graph with mad(G)?10?10/3,then(?) By using Combinatorial Nullstellensatz and discharging method,we studied the neighbor sum distinguishing total coloring of triangle free IC-planar graphs.Main result is described as follow.Let G be a triangle free IC-planar graph,then?"?(G)?max{?(G)+3,10}.