Improper Coloring Of IC-planar Graphs

Graph theory originated from the Konigberg seven bridges problem,this leads to a series of research directions,the theory of graph coloring is an important branch of research,the classical coloring problem has been popularized in many aspects,and the improper coloring is one of them.We consider only simple,finite,undirected and nonempty graphs in this paper.For a graph G=(V,E),we use E=E(G),V=V(G),F=F(G),?(G)and?(G)to denote its edge set,vertex set,face set,minimum degree and maximum degree,respectively.If k?N+,mapping ?:V? {1,2,…,k},if for(?)uv?E,all have?(u)??(v),then we say ? is the point of a normal G dyeing,or simply,graph G is k-colorable.If we represent the partition of vertex sets of graph G in terms of V1,V2,...,Vk,graph G is k-colorable if and only if V1,V2,...,Vk is an independent set.If we relax the condition,we have the concept of improper coloring:Let d1,d2,...,dk be k non-negative integers.A graph G is(d1,d2,...,dk)-colorable,or simply,(d1,d2,...,dk)-colorable,if the vertex set of G can be partitioned into subsets V1,V2,...,Vk such that the subgraph G[Vi]induced by Vi has maximum degree at most di for i=1,2,...,k.In this paper,we mainly improve on previous research results and studies the prob-lems related to the improper coloring of IC-planar graphs.In Chapter 1,we introduce some basic definitions related to graph coloring,at the same time,the notions used in the article are given.Next,we show the definitions and research preliminaries of improper coloring.Finally,we provide the main results of this paper.In Chapter 2,by constructed a minimum counterexample,it is proved that IC-planar graphs with girth at least 6 are(3,0,0)-colorable by discharging method.In Chapter 3,the structural properties of the graph are analyzed,set the initial charge function,it is proved that IC-planar graphs with girth at least 7 are(1,0,0,0)-colorable by discharging method.In Chapter 4,we summarize the main results of this thesis and give the prospects.