Pk+1–Decompositions of Eulerian Graphs: Complexity and Some Solvable Cases

We consider the problem of PMk+1-decomposition of a simple eulerian graph G, that is, decomposition of G into edge disjoint paths of length k. We show that the problem of deciding whether there exists a Pk+1 – decomposition of an eulerian simple graph is NP–complete, for every k ≥ 3. However we find some new classes of graphs where the problem of P4-decomposition can be solved polynomially. We show that an eulerian simple graph G on 3m ≥ 6 edges admits a P4–decomposition if G has no cut vertex v such that exactly one of the components in the graph G − ν has two vertices. In particular, this implies that a 2-connected eulerian simple graph G on 3m ≥ 6 edges admits a P4 –decomposition.