The linear arboricity of K5-minor free graphs

Abstract

The linear arboricity l a ( G ) of a graph G is the minimum number of linear forests which partition the edges of G . In 1980, Akiyama et al. conjectured that for any graph G , ⌈ Δ ( G ) 2 ⌉ ≤ l a ( G ) ≤ ⌈ Δ ( G ) + 1 2 ⌉ . In the paper, we establish two essential structural properties of K 5 -minor free graphs G , one is used to confirm the conjecture for all K 5 -minor free graphs, the other is devoted to proving that l a ( G ) = ⌈ Δ ( G ) 2 ⌉ holds if Δ ( G ) ≥ 9 . In addition, we prove here that if G is a K 5 − -minor free graph and Δ ( G ) ≥ 5 , then l a ( G ) = ⌈ Δ ( G ) 2 ⌉ .