Introduction and definitions

Suppose G1, G2, …, Gκ are graphs of order at most n. We say that there is a packing of G1, G2, …, Gκ into the complete graph κn if there exist injections αi : V(Gi) + V (κn), i = 1,2,…, κ such that α*i(E(Gi)) ∩ α*j(E(Gj)) = Φ for i ≠ j, where the map α*i : E(Gi) → E(kn) is induced by αi. Similarly, suppose G is a graph of order m and H is a graph of order n ≥ m and there exists an injection α : V(G) → V(H) such that α*(E(G)) ∩ E(H) = Φ. Then we say that there is a packing of G into H, and in case n = m, we also say that there is a packing of G and H or G and H are packable. Thus G can be packed into H if and only if G is embeddable in the complement H of H. However, there is a slight difference between embedding and packing. In the study of embedding of a graph into another graph, usually at least one of the graphs is fixed whereas in the study of packing of two graphs very often both the graphs are arbitrarily chosen from certain classes.

In practice, one would like to find an efficient algorithm to pack two graphs G and H. But this has been shown to be an NP-hard problem (see Garey and Johnson [79;p.64]).