A Planar Linear Arboricity Conjecture

TL;DR

This paper advances the understanding of the Linear Arboricity Conjecture for planar graphs, proving it for even maximum degrees at least 10 and providing an efficient partitioning algorithm, with some cases still open.

Contribution

It proves the conjecture for even maximum degrees ≥10 in planar graphs and offers an optimal O(n log n) algorithm for partitioning into linear forests.

Findings

Proved LAC for planar graphs with even Δ ≥ 10.

Provided an O(n log n) algorithm for graph partitioning.

Identified open cases for Δ=6 and Δ=8.

Abstract

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. stated the Linear Arboricity Conjecture (LAC), that the linear arboricity of any simple graph of maximum degree $\Delta$ is either $\lceil \tfrac{\Delta}{2} \rceil$ or $\lceil \tfrac{\Delta+1}{2} \rceil$. In [J. L. Wu. On the linear arboricity of planar graphs. J. Graph Theory, 31:129-134, 1999] and [J. L. Wu and Y. W. Wu. The linear arboricity of planar graphs of maximum degree seven is four. J. Graph Theory, 58(3):210-220, 2008.] it was proven that LAC holds for all planar graphs. LAC implies that for $\Delta$ 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 $\Delta \ge 6$. In this paper we show that it is true for any even $\Delta \ge 10$, leaving open only the cases $\Delta=6, 8$. We present also an O(n log n)-time algorithm for partitioning a planar graph into max{la(G),5} linear forests, which is optimal when $\Delta \ge 9$.