On the linear (n−1)(n−1)-arboricity of Kn(m)Kn(m)

A linear kk-forest of an undirected graph GG is a subgraph of GG whose components are paths with lengths at most kk. The linear kk-arboricity of GG, denoted by lak(G)lak(G), is the minimum number of linear kk-forests needed to partition the edge set E(G)E(G) of GG. In the case where the lengths of paths are not restricted, we then have the linear arboricity of GG, denoted by la(G)la(G). In this paper, we obtain the exact value of the linear (n−1)(n−1)-arboricity of any balanced complete nn-partite graph Kn(m)Kn(m).