Given a connected unit-weighted graph, we study the Maximum k-Cutset Problem, consisting in cutting a graph into k vertex-disjoint sub-graphs, each connected. We propose some exact and heuristic solutions to solve the problem. The first Integer Linear Program is based on a combination of a cut model and an assignment model. The two other Integer Linear Programs are based on the existence of spanning trees. The presented heuristics are clustering algorithms using local search. Experiments have been run on randomly generated instances and a specific set of instances. We observe that the running times are related to the tree-arboricity of graphs. The tree-arboricity is also used to ensure the optimality of solutions found by our heuristics.