Acyclic Chromatic Index Of 1-planar Graphs

A graph is planar if it can be drawn on the plane sueh that every edge isn't crossed.A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge.All planar graphs are 1-planar graphs.Acyclic edge coloring of a graph G is a proper edge coloring such that every cycle is colored with at least three colors.The acyclic chromatic index Xa'(G)of G is the smallest integer k so that G has an acyclic edge coloring using k colors.It was conjectured that Xa'(G)??(G)+ 2 for any graph G with maximum degree ?(G)in 1978.In this paper,we prove that if G is a 1-planar graph without triangle,then Xa'(G)??(G)+ 14.And if G is a 1-planar graph with girth at least 5,then Xa'(G)? ?(G)+ 7.The paper is divided into five chapters.Chapter 1 introduces the origin of acyclic edge coloring concepts and brief introduc-tion of the structure and works in the paper.Chapter 2 relates concepts of this text that will use soome of the basic facts,lists some basic theorems to be used throughout the thesis,and the concepts of acyclic edge coloring,acyclic chromatic index,planar graph and 1-planar graph.Chapter 3 relates some facts and structural results.It is preparing for two chapters proof in behind.Chapter 4 is the first,result.On the basis of known acyclic chromatic index results,combined with main research results and discharge method,we get a new result.That is acyclic edge coloring of 1-planar without triangle is at most ? + 14.Chapter 5 is the second result.On the basis of known acyclic chromatic index results,combined with main research results and discharge method,we prove that acyclic edge coloring of 1-planar with girth at least 5 is at most ? + 7.