Linear 3-Arboricity of the Balanced Complete

A linear \( k \)-forest is a graph whose components are paths with lengths at most \( k \). The minimum number of linear \( k \)-forests needed to decompose a graph \( G \) is the linear \( k \)-arboricity of \( G \) and is denoted by \( la_k(G) \). In this paper, we study the linear \( 3 \)-arboricity of balanced complete multipartite graphs and we obtain some substantial results.