Existence and embeddings of path coverings

In this dissertation, we provide necessary and sufficient conditions for the existence of a set S of 3-paths in 4 complete graph Kn having the property that each 2-path in K n lies in exactly one path in S. These are then used to consider the case n ≡ 3 (mod 4) when no such exact covering is possible, and to solve the problem of covering ( k - 1)-paths with k-paths for all k ≥ 3.;Finally, if a (3, 2)-path covering of a complete graph K n is an ordered pair (V, S) where V is the vertex set of Kn and S is a set of 3-paths in Kn with the property that each 2-path in Kn is a subpath of at most one 3-path in S, then a (3,2)-path covering (V, S) is said to be embedded in the (3,2)-covering (W, T) if V ⊆ W and S ⊆ T.