• Chen, Hong-Yu ;
  • Tan, Xiang ;
  • Wu, Jian-Liang
  • Received : 2010.09.13
  • Published : 2012.01.31


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.


planar graph;linear 2-arboricity;cycle


  1. R. E. L. Aldred and N. C. Wormald, More on the linear k-arboricity of regular graphs, Australas. J. Combin. 18 (1998), 97-104.
  2. J. C. Bermond, J. L. Fouquet, M. Habib, and B. Peroche, On linear k-arboricity, Discrete Math. 52 (1984), no. 2-3, 123-132.
  3. G. J. Chang, Algorithmic aspects of linear k-arboricity, Taiwanese J. Math. 3 (1999), no. 1, 73-81.
  4. G. J. Chang, B. L. Chen, H. L. Fu, and K. C. Huang, Linear k-arboricity on trees, Discrete Appl. Math. 103 (2000), no. 1-3, 281-287.
  5. B. L. Chen, H. L. Fu, and K. C. Huang, Decomposing graphs into forests of paths with size less than three, Australas. J. Combin. 3 (1991), 55-73.
  6. H. L. Fu and K. C. Huang, The linear 2-arboricity of complete bipartite graphs, Ars Combin. 38 (1994), 309-318.
  7. M. Habib and B. Peroche, Some problems about linear arboricity, Discrete Math. 41 (1982), no. 2, 219-220.
  8. B. Jackson and N. C. Wormald, On linear k-arboricity of cubic graphs, Discrete Math. 162 (1996), no. 1-3, 293-297.
  9. K. W. Lih, L. D. Tong, and W. F. Wang, The linear 2-arboricity of planar graphs, Graphs Combin. 19 (2003), no. 2, 241-248.
  10. Q. Ma and J. L. Wu, Planar graphs without 5-cycles or without 6-cycles, Discrete Math. (2008), doi:10.1016/j.disc.2008.07.033.
  11. J. Qian and W. F. Wang, The linear 2-arboricity of planar graphs without 4-cycles, J. Zhejiang Norm. Univ. 29 (2006), no. 2, 121-125.
  12. C. Thomassen, Two-coloring the edges of a cubic graph such that each monochromatic component is a path of length at most 5, J. Combin. Theory Ser. B 75 (1999), no. 1, 100-109.

Cited by

  1. On the linear 2-arboricity of planar graph without normally adjacent 3-cycles and 4-cycles vol.94, pp.5, 2017,