Hong-Yu Chen, Xiang Tan, and Jian-Liang Wu Shanghai Institute of Technology, Shandong University of Finance, Shandong University

Abstract : 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 $G$.

Keywords : planar graph, linear 2-arboricity, cycle