A Near Packing of Two Graphs

Let G1 and G2 be two graphs on n vertices with Δ(G1)=d1 and Δ(G2)=d2. A packing of G1 and G2 is a set L={H1, H2} such that H1≅G1, H2≅G2, and H1 and H2 are edge disjoint subgraphs of Kn. B. Bollobás and S. E. Eldridge (1978, J. Combin. Theory Ser. B25, 105–124) made the following conjecture. If (d1+1)·(d1+1)⩽n+1, then there is a packing of G1 and G2. The degree condition stated in this conjecture could not be weakened, since there exist two such graphs for which (d1+1)·(d2+1)=n+2, and no packing is possible. N. Sauer and J. Spencer (1978, J. Combin. Theory Ser. B25, 295–302) proved that if d1·d2<12n then there is a packing of G1 and G2. In this paper, a generalization and extension of the Sauer–Spencer result is proven. We define a near packing of degree d of two graphs of order n as a generalization of a packing. In a near packing of degree d, the copies of the two graphs may overlap so that the maximum degree of the subgraph defined by the edges common to both copies is d. Thus a near packing of degree 0 is a packing. The result can then be stated as follows. Let a∈R+. If d1·d2⩽an, then there exists a near packing of degree d of G1 and G2 for some d<2a. Furthermore, if (d1+1)·(d2+1)⩽n+1, then the maximum degree d of the subgraph defined by the edges common to both the copy of G1 and the copy of G2 is at most 1 and its size is no larger than n2+1−d1−d2.