Multidesigns for Graph-Pairs of Order 4 and 5

Abstract

The graph decomposition problem is well known. We say a subgraph G divides K _m if the edges of K _m can be partitioned into copies of G. Such a partition is called a G-decomposition or G-design. The graph multidecomposition problem is a variation of the above. By a graph-pair of order t, we mean two non-isomorphic graphs G and H on t non-isolated vertices for which G∪H≅K _t for some integer t≥4. Given a graph-pair (G,H), if the edges of K _m can be partitioned into copies of G and H with at least one copy of G and one copy of H, we say (G,H) divides K _m . We will refer to this partition as a (G,H)-multidecomposition. In this paper, we consider the existence of multidecompositions for several graph-pairs. For the pairs (G,H) which satisfy G∪H≅K ₄ or K ₅, we completely determine the values of m for which K _m admits a (G,H)-multidecomposition. When K _m does not admit a (G,H)-multidecomposition, we instead find a maximum multipacking and a minimum multicovering. A multidesign is a multidecomposition, a maximum multipacking, or a minimum multicovering.