THE LINEAR 2-ARBORICITY OF PLANAR GRAPHS WITHOUT ADJACENT SHORT CYCLES
Hongyu Chen, Xiang Tan, Jianliang Wu · Bulletin of the Korean Mathematical Society · 2012
Let G be a planar graph with maximum degree $\Delta$ . 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. In this paper, we prove that (1) $la_2(G){\leq}{\lceil}\frac{\Delta}{2}\rceil+8$ if G has no adjacent 3-cycles; (2) $la_2(G){\leq}{\lceil}\frac{\Delta}{2}\rceil+10$ if G has no adjacent 4-cycles; (3) $la_2(G){\leq}{\lceil}\frac{\Delta}{2}\rceil+6$ if any 3-cycle is not adjacent to a 4-cycle of G.