A planar linear arboricity conjecture

  • Authors:

  • Marek Cygan;Łukasz Kowalik;Borut Lužar

  • Affiliations:

  • Institute of Informatics, University of Warsaw, Poland;Institute of Informatics, University of Warsaw, Poland;Institute of Mathematics, Physics, and Mechanics, Ljubljana, Slovenia

  • Venue:
  • CIAC'10 Proceedings of the 7th international conference on Algorithms and Complexity
  • Year:
  • 2010

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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.