The Linear $2$-arboricity of Complete Bipartite Graphs

Abstract

A forest in which every component is a path is called a path forest. A family of path forests whose edge sets form a partition of the edge set of a graph $G$ is called a path decomposition of a graph $G$. The minimum number of path forests in a path decomposition of a graph $G$ is the linear arboricity of $G$ and denoted by $\ell (G)$. If we restrict the number of edges in each path to be at most $k$ then we obtain a special decomposition. The minimum number of path forests in this type of decomposition is called the linear $k$-arboricity and denoted by $\ell {\alpha }_k(G)$. In this paper we concentrate on the special type of path decomposition and we obtain the answers for $\ell {\alpha }_2(G)$ when $G$ is $K_{n,n}$. We note here that if we restrict the size to be one, the number $\ell {\alpha }_1(G)$ is just the chromatic index of $G$.