{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:33Z","timestamp":1753893813030,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>A\u00a0packing 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\u00a0 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).<\/jats:p>","DOI":"10.37236\/2998","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T01:32:35Z","timestamp":1578706355000},"source":"Crossref","is-referenced-by-count":2,"title":["Near Packings of Graphs"],"prefix":"10.37236","volume":"20","author":[{"given":"Andrzej","family":"Zak","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2013,5,24]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i2p36\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i2p36\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:20:21Z","timestamp":1579260021000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i2p36"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,5,24]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2013,4,9]]}},"URL":"https:\/\/doi.org\/10.37236\/2998","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2013,5,24]]},"article-number":"P36"}}