Near Packings of Graphs

A packing of a graph G is a set {G1, G2} such that G1 ∼= G, G2 ∼= G, and G1 and G2 are edge disjoint subgraphs of Kn. Let F be a family of graphs. A near packing admitting F of a graph G is a generalization of a packing. In a near packing admitting F , the two copies of G may overlap so the subgraph defined by the edges common to both copies is a member of F . In the paper we study three families of graphs (1) Ek – the family of all graphs with at most k edges, (2) Dk – the family of all graphs with maximum degree at most k, and (3) Ck – the family of all graphs that do not contain a subgraph of connectivity greater than or equal to k + 1. By m(n,F) we denote the maximum number m such that each graph of order n and size less than or equal to m has a near-packing admitting F . It is well known that m(n, C0) = m(n,D0) = m(n, E0) = n − 2 because a near packing admitting C0, D0 or E0 is just a packing. We prove some generalization of this result, namely we prove that m(n, Ck) ≈ (k + 1)n, m(n,D1) ≈ 3 2n, m(n,D2) ≈ 2n. We also present bounds on m(n, Ek). Finally, we prove that each graph of girth at least five has a near packing admitting C1 (i.e. a near packing admitting the family of the acyclic graphs).