On linear 2-arboricity of certain graphs

The linear 2-arboricity of a graph G is the least number of forests which decomposes E(G ) and each forest is a collection of paths of length at most two. A graph has property P k , if each subgraph H satisfies one of the three conditions: (i) delta(H) <= 1 ; (ii) there exists xy is an element of E(H) with deg H (x ) + deg H (y ) <= k ; (iii) H contains a 2-alternating cycle. In this paper, we give two edge-decompositions of graphs with property P k . Using these decompositions, we give an upper bound for the linear 2-arboricity in terms of P k . We also prove that every plane graph with no 12 + -vertex incident with a gem at the center has property P 13 , and graphs with maximum average degree less than 6 k -6 integer.