The linear 2-arboricity of planar graphs without adjacent short cycles
Bull. Korean Math. Soc. 2012 Vol. 49, No. 1, 145-154
Published online January 1, 2012
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
MSC numbers : 05C15
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