Equipartite gregarious connected (5,5)-graph systems

A (p,q)-graph is a graph with p vertices and q edges. There are only five non-isomorphic connected (5,5)-graph: bull graph H1, cricket graph H2, (3,2)-tadpole graph H3, (4,1)-tadpole graph H4, 5-cycle graph H5. Let G be a graph. For i=1,2,3,4,5, an Hi decomposition of G is a pair (X,Bi) where X is the vertex set of G and Bi is a collection of edge-disjoint Hi which partition the edge set of G. Bi is called an Hi system of G. If G is a multipartite graph and the vertices of Hi in Bi are in different partite sets of G, then Bi is a gregarious Hi system of G. Let Kn(m) be a complete n-equi-partite graph with n parts of size m. In this paper, we prove the necessary and sufficient conditions for the existence of a gregarious Hi system of Kn(m), for i=1,2,3,4.