Distinguishing Coloring Of IC-planar Graphs

The theory of graph coloring originated from the Four Color Problem raised by Frederick Guthrie in 1852.The theory of graph coloring is a very important branch of graph theory,and it has a wide range of applications.In recent years,scholars'study on graph coloring is not limited to vertex coloring,edge color-ing and total coloring,on the basis of these colorings,some new colorings were proposed.The adjacent vertex distinguishing edge coloring of a graph is gener-ated by adding constraints on the proper edge coloring of a graph,it requires the proper edge coloring satisfies that for every pair of adjacent vertices u and v,the set of colors used on the edges incident with u differs from the set of colors used on edges incident with v.In 2002,Zhang et al first studied this coloring and put forward a conjecture about the adjacent vertex distinguishing edge coloring,that is,for any connected graph G with order at least 6,?(G)+2 colors can ensure that G has an adjacent vertex distinguishing edge coloring,where ?(G)is the maximum degree of G.And they proved that this conjecture is true for some special graphs.The neighbor sum distinguishing total coloring of a graph is gen-erated by adding constraints on the proper total coloring of a graph,it requires the proper total coloring satisfies that for every pair of adjacent vertices u and v,the sum of colors of u and the edges incident with u differs from the sum of colors of v and the edges incident with v.In 2015,Pilsniak and Wozniak studied this coloring and put forward a conjecture about the neighbor sum distinguishing total coloring,that is,for any graph G,?(G)+3 colors can ensure that G has a neighbor sum distinguishing total coloring,where ?(G)is the maximum degree of G.And they also proved that this conjecture is true for some special graphs.After these two conjectures were proposed,some scholars are devoted to studying whether these conjectures hold for other graphs.This paper studies the above two kinds of colorings of IC-planar graphs.IC-planar graph is a generalization of planar graph,and it is a subclass of 1-planar graph.The study of IC-planar graphs has strong practical significance,it is closely related to circuit layout problems and fire prevention problems.By contradiction,Combinatorial Nullstellensatz,discharging method and so on,this paper proves that the above two conjectures are true for some IC-planar graphs.The main study contents and results are as follows:1.Through the methods of contradiction,we study the structural properties of IC-planar graphs with maximum degree at least 13,and by discharging method,we prove that these graphs satisfy the neighbor sum distinguishing total coloring conjecture.2.Through the methods of Combinatorial Nullstellensat,we study the struc-tural properties of IC-planar graphs with maximum degree at least 10 and without adjacent triangles,and by discharging method,we prove that these graphs satisfy the neighbor sum distinguishing total coloring conjecture.3.Through the methods of recoloring,we study the structural properties of triangle free IC-planar graphs with maximum degree at least 10,and by discharg-ing method,we prove that these graphs satisfy the adjacent vertex distinguishing edge coloring conjecture.