Vertex Shade Degree And Neighbor-distinguishable Edge Coloring Problem Of Special Graph

Colorings theory of graph is an important branch of graph theory.As a simple classification method,colorings of graph has been used in many fields.Colorings of graph is to color the vertices or edges of graph according to certain rules,and divide them into corresponding subsets according to different colors.Vertex arboricity of graph is to divide the vertices of the graph so that the vertices of each subset induces a tree or forest as a subgraph.Chartrand,Kronk and Wall firstly proposed the concept of vertex arboricity,and proved va(G)≤[△(G)+1/2]for every graph.Normal edge colorings refers to a color assignment C:E(G)→ {1,2,…,k} of graph G,such that each edge has a color,and any two adjacent edges have different colors.Let w(u)denote the sum of colors of the edges incident with u.If graph G has a normal edge coloring C:E(G)→{1,2,…,k},such that w(u)#w(v)holds for any two adjacent vertices u,v,then C is said to be a neighbor sum distinguishing edge colorings of graph G.In this paper,we mainly study the vertex arboricity of planar graphs under restricted conditions,and the neighbor sum distinguishing edges colorings of graphs under restricted conditions.Firstly,we mainly use the method of contradiction and discharging method to prove va(G)≤ 2 for every planar graphs without intersecting 4-cycles and 6-cycles.Next,we mainly use the method of contradiction,Combinatorial Nullstellensatz and discharging method to prove that if the graph G is a graph without isolated edges,and mad(G)<3,△(G)≥ 6,then χ’∑(G)≤△(G)+1.Finally,through the solutions to these problems that we have mastered and the existing conclusions,we make a summary and outlook on the Vertex arboricity of planar graphs and the neighbor sum distinguishing edge coloring of graphs.