Summary: Linear arboricity and linear k-arboricity of regular graphs
V. J. Teague
N. C. Wormald
We find upper bounds on the linear k-arboricity of d-regular graphs using a proba-
bilistic argument. For small k these bounds are new. For large k they blend into the
known upper bounds on the linear arboricity of regular graphs.
A linear forest is a forest each of whose components is a path. The linear arboricity of a
graph G is the minimum number of linear forests required to partition E(G) and is denoted
by la(G). It was shown by Akiyama, Exoo and Harary  that la(G) = 2 when G is cubic,
and they conjectured that every d-regular graph has linear arboricity exactly (d + 1)/2 .
This was shown to be asymptotically correct as d in , and in  the following result
Theorem 1 There is an absolute constant c > 0 such that for every d-regular graph G