Almost every graph can be covered by d ∆ 2 e linear forests
نویسندگان
چکیده
A linear forest is the union of a set of vertex disjoint paths. Akiyama, Exoo and Harary and independently Hilton have conjectured that the edges of every graph of maximum degree ∆ can be covered by d∆+1 2 e linear forests. We show that almost every graph can be covered with this number of linear forests. 1 Linear Forests, Directed Cycles, and the Probabilistic Method A linear forest is the union of a set of vertex disjoint paths. For a graph G, the linear arboricity of G, denoted by la(G) (and defined by Harary in [13]), is the least number of linear forests required in a covering (or equivalently a partitioning) of its edge set E(G). Clearly, if G has maximum degree ∆ then
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