 
Summary: ISRAELJOURNALOFMATHEMATICS,Vol.62,No.3,1988
THE LINEAR ARBORICITY OF GRAPHS
BY
N. ALONt
Department ofMathematics, SacklerFacultyofExact Sciences,
TelAviv University,Ramat Aviv, TelAviv, Israel
ABSTRACT
A linearforest is a forest in which each connected component is a path. The
lineararboricityla(G) of a graph G is the minimum number of linear forests
whose union is the set ofall edges of G. The lineararboricityconjectureasserts
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) _<(½+ e)Afor every Gwith maximum degree A >_Ao.To do this, we first
prove the conjecture for every G with an even maximum degree A and with
girthg > 50A.
1. Introduction
All graphs considered here are finite, undirected and simple, i.e., have no
