A linear forest is a union of vertex-disjoint paths, and the linear
arboricity of a graph G, denoted by la(G), is the minimum
number of linear forests needed to partition the edge set of G. Clearly,
la(G)β₯βΞ(G)/2β for a graph G with maximum
degree Ξ(G). On the other hand, the Linear Arboricity Conjecture due to
Akiyama, Exoo, and Harary from 1981 asserts that la(G)β€β(Ξ(G)+1)/2β for every graph G. This conjecture has been
verified for planar graphs and graphs whose maximum degree is at most 6, or
is equal to 8 or 10.
Given a positive integer k, a graph G is k-degenerate if it can be
reduced to a trivial graph by successive removal of vertices with degree at
most k. We prove that for any k-degenerate graph G, la(G)=βΞ(G)/2β provided Ξ(G)β₯2k2βk.Comment: 15 pages, 1 figur