A Planar Linear Arboricity Conjecture

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.

Supported in part by bilateral project BI-PL/08-09-008. M. Cygan and Ł. Kowalik were supported in part by Polish Ministry of Science and Higher Education grant N206 355636.