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The linear arboricity of graphs. (English) Zbl 0673.05019
Summary: A linear forest is a forest in which each connected component is a path. The linear arboricity $$\ell a(G)$$ 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 $$\Delta =\Delta (G)$$, $\ell a(G)\leq \lceil (\Delta +1)/2\rceil.$ Although this conjecture received a considerable amount of attention, it has been proved only for $$\Delta\leq 6$$, $$\Delta =8$$ and $$\Delta =10$$, and the best known general upper bound for $$\ell a(G)$$ is $$\ell a(G)\leq \lceil 3\Delta /5\rceil$$ for even $$\Delta$$ and $$\ell a(G)\leq \lceil (3\Delta +2)/5\rceil$$ for odd $$\Delta$$. Here we prove that for every $$\epsilon >0$$ there is a $$\Delta_ 0=\Delta_ 0(\epsilon)$$ so that $$\ell a(G)\leq (+\epsilon)\Delta$$ for every G with maximum degree $$\Delta \geq \Delta_ 0$$. To do this, we first prove the conjecture for every G with an even maximum degree $$\Delta$$ and with girth $$g\geq 50\Delta$$.

##### MSC:
 05C05 Trees 05C38 Paths and cycles 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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