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- W1546647622 abstract "A packing of a graph $G$ is a set ${G_1,G_2}$ such that $G_1cong G$, $G_2cong G$, 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)." @default.
- W1546647622 created "2016-06-24" @default.
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- W1546647622 date "2013-05-24" @default.
- W1546647622 modified "2023-10-18" @default.
- W1546647622 title "Near Packings of Graphs" @default.
- W1546647622 doi "https://doi.org/10.37236/2998" @default.
- W1546647622 hasPublicationYear "2013" @default.
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