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A Planar Linear Arboricity Conjecture

  • Marek Cygan
  • Łukasz Kowalik
  • Borut Lužar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6078)

Abstract

The linear arboricity la(G) of a graph G is the minimum number of linear forests (graphs where every connected component is a path) that partition the edges of G. In 1984, Akiyama et al. [1] stated the Linear Arboricity Conjecture (LAC), that the linear arboricity of any simple graph of maximum degree Δ is either \(\big \lceil \tfrac{\Delta}{2} \big \rceil\) or \(\big \lceil \tfrac{\Delta+1}{2} \big \rceil\). In [14,15] it was proven that LAC holds for all planar graphs.

LAC implies that for Δ odd, \({\rm la}(G)=\big \lceil \tfrac{\Delta}{2} \big \rceil\). We conjecture that for planar graphs this equality is true also for any even Δ ≥ 6. In this paper we show that it is true for any Δ ≥ 10, leaving open only the cases Δ= 6, 8.

We present also an O(nlogn) algorithm for partitioning a planar graph into max {la(G), 5} linear forests, which is optimal when Δ ≥ 9.

Keywords

Planar Graph Maximum Degree Regular Graph Initial Charge Common Neighbor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Akiyama, J., Exoo, G., Harary, F.: Covering and packing in graphs III: Cyclic and acyclic invariants. Math. Slovaca. 30, 405–417 (1980)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Akiyama, J., Exoo, G., Harary, F.: Covering and packing in graphs IV: Linear arboricity. Networks 11, 69–72 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alon, N.: The linear arboricity of graphs. Israel Journal of Mathematics 62(3), 311–325 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Alon, N., Teague, V., Wormald, N.C.: Linear arboricity and linear k-arboricity of regular graphs. Graphs and Combinatorics 17(1), 11–16 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cole, R., Kowalik, Ł.: New linear-time algorithms for edge-coloring planar graphs. Algorithmica 50(3), 351–368 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cole, R., Kowalik, Ł., Škrekovski, R.: A generalization of Kotzig’s theorem and its application. SIAM Journal on Discrete Mathematics 21(1), 93–106 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cygan, M., Kowalik, Ł., Lužar, B.: A planar linear arboricity conjecture. arXiv.org e-Print archive, arXiv:0912.5528v1 (2009)Google Scholar
  8. 8.
    Enomoto, H., Péroche, B.: The linear arboricity of some regular graphs. J. Graph Theory 8, 309–324 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Guldan, F.: The linear arboricity of 10-regular graphs. Math. Slovaca. 36(3), 225–228 (1986)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Harary, F.: Covering and packing in graphs I. Ann. N.Y. Acad. Sci. 175, 198–205 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Henzinger, M.R., King, V.: Randomized fully dynamic graph algorithms with polylogarithmic time per operation. J. ACM 46(4), 502–516 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Peroche, B.: Complexity of the linear arboricity of a graph (In French). RAIRO Oper. Res. 16, 125–129 (1982) (in French)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Vizing, V.G.: Critical graphs with a given chromatic number. Diskret. Analiz 5, 9–17 (1965)Google Scholar
  14. 14.
    Wu, J.L.: On the linear arboricity of planar graphs. J. Graph Theory 31, 129–134 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Wu, J.L., Wu, Y.W.: The linear arboricity of planar graphs of maximum degree seven is four. J. Graph Theory 58(3), 210–220 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wu, J.L., Hou, J.F., Liu, G.Z.: The linear arboricity of planar graphs with no short cycles. Theor. Comput. Sci. 381(1-3), 230–233 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Wu, J.L., Hou, J.F., Sun, X.Y.: A note on the linear arboricity of planar graphs without 4-cycles. In: International Symposium on Operations Research and Its Applications, pp. 174–178 (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Marek Cygan
    • 1
  • Łukasz Kowalik
    • 1
  • Borut Lužar
    • 2
  1. 1.Institute of InformaticsUniversity of WarsawPoland
  2. 2.Institute of Mathematics, Physics, and MechanicsLjubljanaSlovenia

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