Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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THE LINEAR 2-ARBORICITY OF PLANAR GRAPHS WITHOUT ADJACENT SHORT CYCLES

  • Chen, Hong-Yu (School of Sciences Shanghai Institute of Technology) ;
  • Tan, Xiang (School of Statistics and Mathematics Shandong University of Finance) ;
  • Wu, Jian-Liang (School of Mathematics Shandong University)
  • Received : 2010.09.13
  • Published : 2012.01.31
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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.

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References

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  1. On the linear 2-arboricity of planar graph without normally adjacent 3-cycles and 4-cycles vol.94, pp.5, 2017, https://doi.org/10.1080/00207160.2016.1158813