A Planar Linear Arboricity Conjecture

The linear arboricity la(G) of a graph G is the minimum number of linear forests 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 is either 2 or +1 2 . In [14, 17] it was proven that LAC holds for all planar graphs. LAC implies that for odd, la(G) = 2 . We conjecture that for planar graphs this equality is true also for any even 6. In this paper we show that it is true for any even 10, leaving open only the cases = 6, 8. We present also an O(n log n) algorithm for partitioning a planar graph into max{la(G), 5} linear forests, which is optimal when 9.

• Type: Conference
• Year: 2010
• Where: CIAC
• Authors: Marek Cygan, Lukasz Kowalik, Borut Luzar
• Keywords: Algorithms, CIAC 2010, Linear Arboricity Conjecture, Linear Forests, Planar Graphs