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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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