 
Summary: Linear arboricity and linear karboricity of regular graphs
N. Alon
V. J. Teague
N. C. Wormald
Abstract
We find upper bounds on the linear karboricity of dregular 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.
1 Introduction
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 [1] that la(G) = 2 when G is cubic,
and they conjectured that every dregular graph has linear arboricity exactly (d + 1)/2 .
This was shown to be asymptotically correct as d in [3], and in [4] the following result
is shown.
Theorem 1 There is an absolute constant c > 0 such that for every dregular graph G
la(G)
d
2
+ cd2/3
