Randomly -decomposable graphs

A graph G is H -decomposable if it can be expressed as an edge-disjoint union of subgraphs, each subgraph isomorphic to H . If G has the additional property that every H -decomposable subgraph of G is part of an H -decomposition of G , then G is randomly  H -decomposable. Using computer assistance, we provide in this paper a characterization of randomly path-decomposable graphs for paths of length 11 or less. We also prove the following two results: (1) With one small exception, randomly P k -decomposable graphs with a vertex of odd degree do not contain a P k + 1 -subgraph, (2) When the edges of a P k -subgraph are deleted from a connected randomly P k -decomposable graph, the resulting graph has at most one nontrivial component.