THE LINEAR 2-ARBORICITY OF PLANAR GRAPHS WITHOUT ADJACENT SHORT CYCLES

Abstract. Let G be a planar graph with maximum degree ∆. The linear 2-arboricity la2(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. In this paper, we prove that (1) la2(G) ⌈∆2 ⌉+8 if G has no adjacent 3-cycles; (2) la2(G) ⌈∆2 ⌉ + 10 if G has no adjacent 4-cycles; (3) la2(G) ⌈∆2 ⌉+ 6 if any 3-cycle is not adjacent to a 4-cycle of G. 1