On linear 2-arboricity of certain graphs

Abstract

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 Pk, if each subgraph H satisfies one of the three conditions: (i) δ(H)≤1; (ii) there exists xy∈E(H) with degH(x)+degH(y)≤k; (iii) H contains a 2-alternating cycle. In this paper, we give two edge-decompositions of graphs with property Pk. Using these decompositions, we give an upper bound for the linear 2-arboricity in terms of Pk. We also prove that every plane graph with no 12+-vertex incident with a gem at the center has property P13, and graphs with maximum average degree less than 6k−6k+3 have property Pk, where k≥5 is an integer.