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The Linear Arboricity of Planar Graphs without 5-, 6-Cycles with Chords

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Abstract

The linear arboricity la(G) of a graph G is the minimum number of linear forests which partition the edges of G. In this paper, it is proved that for a planar graph G, \({la(G)=\lceil\frac{\Delta(G)}{2}\rceil}\) if Δ(G) ≥ 7 and G has no 5-cycles with chords, or Δ(G) ≥ 5 and G has no 5-, 6-cycles with chords.

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References

  1. Akiyama J., Exoo G., Harary F.: Covering and packing in graphs III: cyclic and acyclic invariants. Math. Slovaca 30, 405–417 (1980)

    MathSciNet  MATH  Google Scholar 

  2. Akiyama J., Exoo G., Harary F.: Covering and packing in graphs IV: linear arboricity. Networks 11, 69–72 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bondy J.A., Murty U.S.R.: Graph Theory with Applications. Macmillan, New York (1976)

    MATH  Google Scholar 

  4. Enomoto H., Péroche B.: The linear arboricity of some regular graphs. J. Graph Theory 8, 309–324 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  5. Guldan F.: The linear arboricity of 10 regular graphs. Math. Slovaca 36, 225–228 (1986)

    MathSciNet  MATH  Google Scholar 

  6. Harary F.: Covering and packing in graphs I. Ann. N.Y. Acad. Sci. 175, 198–205 (1970)

    MATH  Google Scholar 

  7. Wu J.L.: Some path decompositions of Halin graphs. J. Shandong Mining Institute 17, 92–96 (1998)

    Google Scholar 

  8. Wu J.L.: On the linear arboricity of planar graphs. J. Graph theory 31, 129–134 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Wu J.L.: The linear arboricity of series-parallel graphs. Graphs Combin. 16, 367–372 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wu J.L., Hou J.F., Liu G.Z.: The linear arboricity of planar graphs with no short cycles. Theor. Comput. Sci. 381, 230–233 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Wu J.L., Liu G.Z., Wu Y.L.: The linear arboricity of composition of two graphs. J. Syst. Sci. Complex 15, 372–375 (2002)

    MathSciNet  MATH  Google Scholar 

  12. Wu J.L., Wu Y.W.: The linear arboricity of planar graphs of maximum degree seven are four. J. Graph Theory 58, 210–220 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Wu J.L., Hou J.F., Sun X.Y.: A note on the linear arboricity of planar graphs without 4-cycles. ISORA’ Lect. Notes Oper. Res. 10, 174–178 (2009)

    Google Scholar 

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Authors and Affiliations

  1. School of Science, Shanghai Institute of Technology, Shanghai, 201418, People’s Republic of China

    Hongyu Chen

  2. School of Statistics and Mathematics, Shandong University of Finance, Jinan, 250014, Shandong, People’s Republic of China

    Xiang Tan

  3. School of Mathematics, Shandong University, Jinan, 250100, Shandong, People’s Republic of China

    Jianliang Wu & Guojun Li

Authors
  1. Hongyu Chen
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  2. Xiang Tan
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  3. Jianliang Wu
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  4. Guojun Li
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Corresponding author

Correspondence to Jianliang Wu.

Additional information

This work was partially supported by National Natural Science Foundation of China (10971121, 60373025, 60873207).

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Chen, H., Tan, X., Wu, J. et al. The Linear Arboricity of Planar Graphs without 5-, 6-Cycles with Chords. Graphs and Combinatorics 29, 373–385 (2013). https://doi.org/10.1007/s00373-011-1118-y

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  • DOI: https://doi.org/10.1007/s00373-011-1118-y

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