Packing of graphs - a survey

Suppose Gi, G2, . . . , Gk are graphs of order at most n. We say that there is a packing of Gi, G2, . . . , G, into the complete graph K,, if there exist injections q : V(G,)+ V(K,), i = 1, 2, . . . , k such that aT(E(Gi)) fl (Y/*(E(G~)) = $ for i #j, where the map ar: : E(G,)-+ E(K,) . IS induced by a;. Similarly, suppose G is a graph of order m and H is a graph of order n 2 m and there exists an injection (Y: V(G)+ V(H) such that a*(E(G)) fl E(H) = $, 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 d 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 two graphs is fixed whereas in the study of packing of two graphs very often both the graphs are arbitrarily chosen from certain classes of graphs. In this paper we shall concentrate mainly on some results on the following t\vo packing problems. The first one is on dense packings of trees of different sizes into K,,. The second one is on general packings of two graphs having small size. Some open problems in this area will also be mentioned. We shall use the following notation and definitions. The order, the size and the maximum valency of a graph G are denoted by (GI, e(G) and A(G) respectively A tree, a star and a path of order i are denoted respectively by z, Si and 4. The tree obtained from S,_, (n 2 5) by inserting a new vertex on an edge is denoted by SA. The tree obtained from SA_,(n 3 6) by inserting a new vertex on the edge which is not incident with the vertex of maximum valency of S,‘_, is denoted by Sz. We denote by S(6) the tree obtained from S, by adding two new vertices each of which is joined to one end-vertex of S,. The cycle of length m and the null graph of order r are denoted by C, and 0, respectively. The complete bipartite graph having bipartition VI and V2 such that 1 VII = m and IV,! = n is denoted by K The graph obtained from C4 by adding an edge joining two opposite ve?tices is denoted by C,‘. The disjoint union of two graphs G and H is denoted by G U H, and mG stands for the disjoint union of m copies of G. A graph of