The linear arboricity of graphs

A linear forest is a forest in which each connected component is a path. The linear arboricity la(G) of a graph G is the minimum number of linear forests whose union is the set of all edges of G. The linear arboricity conjecture asserts that for every simple graph G with maximum degree Δ=Δ(G), {Mathematical expression}. Although this conjecture received a considerable amount of attention, it has been proved only for Δ≦6, Δ=8 and Δ=10, and the best known general upper bound for la(G) is la(G)≦⌈3Δ/5⌉ for even Δ and la(G)≦⌈(3Δ+2)/5⌉ for odd Δ. Here we prove that for every e{open}>0 there is a Δ₀=Δ₀(e{open}) so that la(G)≦(1/2+e{open})Δ for every G with maximum degree Δ≧Δ₀. To do this, we first prove the conjecture for every G with an even maximum degree Δ and with girth g≧50Δ.