The linear 3-arboricity of Kn,nKn,n and KnKn

A linear k-forest is a forest whose components are paths of length at most k. The linear k-arboricity of a graph G  , denoted by lak(G)lak(G), is the least number of linear k-forests needed to decompose G  . In this paper, we completely determine lak(G)lak(G) when G   is a balanced complete bipartite graph Kn,nKn,n or a complete graph KnKn, and k=3k=3.