Israel Journal of Mathematics

, Volume 62, Issue 3, pp 311–325

# The linear arboricity of graphs

Article

## Abstract

Alinear forest is a forest in which each connected component is a path. Thelinear arboricity la(G) of a graphG is the minimum number of linear forests whose union is the set of all edges ofG. Thelinear arboricity conjecture asserts that for every simple graphG with maximum degree Δ=Δ(G),$$la(G) \leqq [\frac{{\Delta + 1}}{2}].$$. 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 everyɛ>0 there is a Δ00(ɛ) so that la(G)≦(1/2+ɛ)Δ for everyG with maximum degree Δ≧Δ0. To do this, we first prove the conjecture for everyG with an even maximum degree Δ and withgirth g≧50Δ.

## Keywords

Maximum Degree Pairwise Disjoint Regular Graph Dependency Graph Span Subgraph
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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