A Planar Linear Arboricity Conjecture

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


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.


Planar Graph Maximum Degree Regular Graph Initial Charge Common Neighbor 
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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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