### Abstract

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 Δ_{0}=Δ_{0}(e{open}) so that la(G)≦(1/2+e{open})Δ for every G with maximum degree Δ≧Δ_{0}. To do this, we first prove the conjecture for every G with an even maximum degree Δ and with girth g≧50Δ.

Original language | English (US) |
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Pages (from-to) | 311-325 |

Number of pages | 15 |

Journal | Israel Journal of Mathematics |

Volume | 62 |

Issue number | 3 |

DOIs | |

State | Published - Oct 1 1988 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Israel Journal of Mathematics*,

*62*(3), 311-325. https://doi.org/10.1007/BF02783300