Homogeneous factorisations of complete multipartite graphs (Q864129)

Given a graph \(X\), each edge \(uv\) of \(X\) gives rise to two arcs, namely, \((u,v)\) and \((v,u)\). Let \(\mathcal{P}=\{P_1,P_2,\dots,P_k\}\) be a partition of the arc set of \(X\). If there exist subgroups \(M,G\) of \(\mathrm{Aut}(X)\) such that both act transitively on the vertex set of \(X\), \(M\) fixes each \(P_i\) setwise, the partition \(\mathcal{P}\) is invariant under \(G\), and the induced action of \(G\) on \(\mathcal{P}\) is transitive, then we call \((M,G,X,\mathcal{P})\) a homogeneous factorization of \(X\) of index \(k\). This paper concentrates on homogeneous factorizations of complete uniform multipartite graphs and establishes many interesting results and examples.