@MISC{Alon88thelinear, author = {N. Alon}, title = {THE LINEAR ARBORICITY OF GRAPHS }, year = {1988} }

Share

OpenURL

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 A = A(G), Although this conjecture received a considerable amount of attention, it has been proved only for A _-< 6, A = 8 and A = 10, and the best known general upper bound for la(G) is la(G) _-< [3A/5] for even A and la(G) _-< [(3A + 2)/5] for odd A. Here we prove that for every t> 0 there is a Ao = Ao(e) so that la(G) _-< ( + e)A for every G with maximum degree A>_- Ao. To do this, we first prove the conjecture for every G with an even maximum degree A and with girth g> 50A.