Linear k-arboricities on trees

For a fixed positive integer k, the linear k-arboricity lak(G) of a graph G is the minimum number ℓ such that the edge set E(G) can be partitioned into ℓ disjoint sets and that each induces a subgraph whose components are paths of lengths at most k. This paper studies linear k-arboricity from an algorithmic point of view. In particular, we present a linear-time algorithm to determine whether a tree T has lak(T)⩽m.