A Near Packing of Two Graphs

Abstract

Let G 1 and G 2 be two graphs on n vertices with Δ ( G 1 )= d 1 and Δ ( G 2 )= d 2 . A packing of G 1 and G 2 is a set L ={ H 1 , H 2 } such that H 1 ≅ G 1 , H 2 ≅ G 2 , and H 1 and H 2 are edge disjoint subgraphs of K n . B. Bollobás and S. E. Eldridge (1978, J. Combin. Theory Ser. B 25 , 105–124) made the following conjecture. If ( d 1 +1)·( d 1 +1)⩽ n +1, then there is a packing of G 1 and G 2 . The degree condition stated in this conjecture could not be weakened, since there exist two such graphs for which ( d 1 +1)·( d 2 +1)= n +2, and no packing is possible. N. Sauer and J. Spencer (1978, J. Combin. Theory Ser. B 25 , 295–302) proved that if d 1 · d 2 < 1 2 n then there is a packing of G 1 and G 2 . 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 d 1 · d 2 ⩽ an , then there exists a near packing of degree d of G 1 and G 2 for some d <2 a . Furthermore, if ( d 1 +1)·( d 2 +1)⩽ n +1, then the maximum degree d of the subgraph defined by the edges common to both the copy of G 1 and the copy of G 2 is at most 1 and its size is no larger than n 2 +1− d 1 − d 2 .