The Linear 2-arboricity Of A Kind Of Spare Graph And The Linear Arboricity Of 1-planar Graphs

Graph coloring problem is an important branch of the edge coloring of graphs,and it has very practical application in our actual life.The linear arboricity and the linear k-arboricity of a graph G as a class of edge coloring of graphs,has important research significance in coloring and decomposition of graphs.Let G=（V（G）,E（G））be a finite,undirected and simple graph.A linear 2-forest is a graph whose components are paths of length at most 2.The linear 2-arboricity of a graph G is the least integer m such that G can be edge-partitioned into m linear 2-forests,denoted by la2（G）.A linear forest is a graph whose com-ponents are paths.The linear arboricity of a graph G is the minimum number of linear forests which partition the edges of G,denoted by la（G）.In this pa-per,we mainly study the linear 2-arborcity of a graph with mad（G）≤4 and the linear arboricity of 1-planar graphs with Δ（G）≥ 28 and get the following conclusions:1.If G is a graph with mad（G）≤4,then la2（G）≤[Δ（G）/2]+4 ifΔ（G）≡0,3（mod 4）and la2（G）≤[Δ（G）/2]+5 if △（G）≡ 1,2（mod 4）;2.If G is a 1-planar graphs with Δ（G）≥ 28,then la₂（G）=[Δ（G）/2].