Linear Arboricity of 1-Planar Graphs

Abstract

Abstract The linear arboricity la(G) of a graph G is the minimum number of linear forests that partition the edges of G. In 1981, Akiyama, Exoo and Harary conjectured that ⌈ Δ(G)2 ⌉≤la(G)≤⌈ Δ(G)+12 ⌉ \left\lceil {{{\Delta \left( G \right)} \over 2}} \right\rceil \le la\left( G \right) \le \left\lceil {{{\Delta \left( G \right) + 1} \over 2}} \right\rceil for any simple graph G. A graph G is 1-planar if it can be drawn in the plane so that each edge has at most one crossing. In this paper, we confirm the conjecture for 1-planar graphs G with Δ(G) ≥ 13.