The Linear 2-arboricity Of 1-planar Graphs

The linear k-arboricity of a graph G,denoted by lak(G),is the least integer m such that G can be edge-partitioned into m linear k-forests.Clearly,lak(G)? lak+1(G)for any k?1.For extremities,la1(G)is the chromatic index ?'(G)of G,and la?(G)is the linear arboricity la(G)of G.The linear k-arboricity of a graph was introduced by Habib and Peroche in 1982.Among other things,the following challenging problem was proposed:Let G be a graph with n vertices and k? 2 be an integer.ThenIn this thesis,we study the linear 2-arboricity of 1-planar graph.The thesis consists of four chapters.In Chapter 1,we first introduced the basic concepts and symbols involved in this thesis,then briefly described the research status in related fields,and finally presented the main results of this thesis.Meanwhile,we stated some lemmas on properties of 1-planar graphs and edge-partition theorems of planar graphs.In Chapter 2,we study the 1-planar graphs without 3-cycles.It is showed that for any 1-planar graph G without 3-cycles:(1)if ?(G)? 2,then G contains a 21-light edge or a 2-alternating-cycle;(2)la2(G)?[?+1/2]+ 6.In Chapter 3,we study the 1-planar graphs without 4-cycles.It is showed that for any 1-planar graph G without 4-cycles:(1)if ?(G)? 2,then G contains a 16-light edge or a 2-alternating-cycle;(2)la2(G)?[?+1/2]+6.In Chapter 4,we consider 1-planar graphs with high girth.It is showed that for any 1-planar graph G:(1)if g(G)? 5 and ?(G)? 2,then G contains a 12-light edge or a 2-alternating-cycle;(2)if g(G)? 6 and ?(G)? 2,then G contains a 11-light edge or a 2-alternating-cycle;(3)if g(G)?6,then la2(G)?[?+1/2]+5.