The linear 2-arboricity of planar graphs

Let G be a planar graph with maximum degree Delta and girth g. The linear 2-arboricity la(2)(G) of G is the least integer k such that G can be partitioned into k edge-disjoint forests, whose component trees are paths of length at most 2. We prove that (1) la(2)(G)less than or equal toinverted right perpendicular(Delta+1)/2inverted left perpendicular+12; (2) la(2)(G)less than or equal toinverted right perpendicular(Delta+1)/2inverted left perpendicular+6 if ggreater than or equal to4; (3) la(2)(G)less than or equal toinverted right perpendicular(Delta+1)/2inverted left perpendicular+2 if ggreater than or equal to5; (4) la(2)(G)less than or equal toinverted right perpendicular(Delta+1)/2inverted left perpendicular+1 if ggreater than or equal to7.