Near Packings of Graphs

Abstract

A packing of a graph $G$ is a set $\{G_1,G_2\}$ such that $G_1\cong G$, $G_2\cong G$, and $G_1$ and $G_2$ are edge disjoint subgraphs of $K_n$. Let $\mathcal{F}$ be a family of graphs. A near packing admitting $\mathcal{F}$ of a graph $G$ is a generalization of a packing. In a near packing admitting $\mathcal{F}$, the two copies of $G$ may overlap so the subgraph defined by the edges common to both copies is a member of $\mathcal{F}$. In the paper we study three families of graphs (1) $\mathcal{E}_k$ -- the family of all graphs with at most $k$ edges, (2) $\mathcal{D}_k$ -- the family of all graphs with maximum degree at most $k$, and (3) $\mathcal{C}_k$ -- the family of all graphs that do not contain a subgraph of connectivity greater than or equal to $k+1$. By $m(n,\mathcal{F})$ 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 $\mathcal{F}$. It is well known that $m(n,\mathcal{C}_0)=m(n,\mathcal{D}_0)=m(n,\mathcal{E}_0)=n-2$ because a near packing admitting $\mathcal{C}_0$, $\mathcal{D}_0$ or $\mathcal{E}_0$ is just a packing. We prove some generalization of this result, namely we prove that $ m(n,\mathcal{C}_k)\approx (k+1)n$, $ m(n,\mathcal{D}_1)\approx \frac{3}{2}n$, $ m(n,\mathcal{D}_2)\approx 2n$. We also present bounds on $m(n,\mathcal{E}_k)$. Finally, we prove that each graph of girth at least five has a near packing admitting $\mathcal{C}_1$ (i.e. a near packing admitting the family of the acyclic graphs).