Multidesigns for Graph-Triples of Order 6

We call T = (G1, G2, G3) a graph-triple of order t if the Gi are pairwise non-isomorphic graphs on t non-isolated vertices whose edges can be combined to form Kt. If m ≥ t, we say T divides Km if E(Km) can be partitioned into copies of the graphs in T with each Gi used at least once, and we call such a partition a T-multidecomposition. In this paper, we study multidecompositions of Km for graph-triples of order 6. We focus on graph-triples in which either one graph is a perfect matching or all graphs have 5 edges each. Moreover, we determine maximum multipackings and minimum multicoverings when Km does not admit a multidecomposition.