Some structural results on linear arboricity

A linear forest-factor F of a graph G is a spanning subgraph of G whose components are paths. A linear forest-decomposition of G is a collection :F = {F1, ••• , Fk } of linear forest-factors of G such that the edge set E (G) of G is the disjoint union of E (F1), •.• , E (Fk)' The linear ar borici ty la( G) of G is the minimum cardinality of a linear forest-decomposition of G. In this paper we evolve a method to construct a small linear forestdecomposition of a graph G from given linear forest-decompositions of two subgraphs that are linked by a cut vertex of G. As an application we determine the linear arboricity of block-cactus graphs which extends a result of Zelinka [5] (1986). Our results are connected to the "linear arboricity conjecture" of Akiyama, Exoo and Harary [2] (1980).