2-distance Coloring Of Planar Graphs

All graphs are finite simple graphs in this paper.For a graph G,we use V(G),E(G),F(G),?(G),?(G),and g(G)to denote the vertex set,edge set,face set,maximum degree,minimum degree and girth,respectively.A graph G is Planar if it can be embedded in a plane and its edges intersect only at its endpoints.A planar graph is called a planar graph if it can be embedded in a plane.The vertices and edges of a plane graph G divide the whole plane into connected regions whose closures are called planes of the plane graph.The k-2-distance coloring of graph G is a mapping ?:V(G)? {1,2,…,k}if any two vertices at distance at most two from each other get different colors.The 2-distance chromatic number is the smallest integer k such that G has a k-2-distance coloring,denoted by ?2(G).In 1977,Wegner proposed the conjecture:for planar graph G,(1)if ?(G)=3,then ?2(G)?7;(2)if 4 ??(G)?7,then ?2(G)??(G)+5;(3)if ?(G)>8,then?2(G)?[3?(G)/2]+1.According this conjecture,many scholars at home and abroad have carried out research,but this conjecture is still open.This dissertation studies some conclusions about 2-distance coloring of planar graphs.In the first chapter,we introduced some basic concepts and research status.In the second chapter,we study the plane graph G that lack of some certain short cycles with ?2(G)??(G)+3.In the third chapter,we study the plane graph G without intersecting 5-cycle and 6--cycle,if ?(G)? 31,g(G)? 5,then ?2(G)? ?(G)+2.In this paper,we use the method of counter proof to analyze the structural properties of planar graphs without certain short cycles through minimal counter examples,and then use Euler formula and weight transfer to derive contradictions.So as to get the desired conclusion.