On the linear k-arboricity of Kn and Kn,n Original Research Article

A linear k-forest of a undirected graph G is a subgraph of G whose components are paths with lengths at most k. The linear k-arboricity of G, denoted by lak(G), is the minimum number of linear k-forests needed to partition the edge set E(G) of G. In case that the lengths of paths are not restricted, we then have the linear arboricity of G, denoted by la(G). In this paper, we first prove that a conjecture by Habib and Peroche holds when G is Kn or Kn,n and k is not less than half the order. Secondly, I(G)=min{k|lak(G)=la(G)} is determined for G is Kn or Kn,n.