Decompositions into two linear forests of bounded lengths

Abstract: For some $k \in \mathbb{Z}{\geq 0}\cup \infty$, we call a linear forest $k$-bounded if each of its components has at most $k$ edges. We will say a $(k,\ell)$-bounded linear forest decomposition of a graph $G$ is a partition of $E(G)$ into the edge sets of two linear forests $F_k,F\ell$ where $F_k$ is $k$-bounded and $F_\ell$ is $\ell$-bounded. We show that the problem of deciding whether a given graph has such a decomposition is NP-complete if both $k$ and $\ell$ are at least $2$, NP-complete if $k\geq 9$ and $\ell =1$, and is in P for $(k,\ell)=(2,1)$. Before this, the only known NP-complete cases were the $(2,2)$ and $(3,3)$ cases. Our hardness result answers a question of Bermond et al. from 1984. We also show that planar graphs of girth at least nine decompose into a linear forest and a matching, which in particular is stronger than $3$-edge-colouring such graphs.