{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:03Z","timestamp":1753893783929,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We define two notions of generation between the various optimal packings ${\\cal Q}_m^K$ of $m$ congruent disks in a subset $K$ of ${\\Bbb R}^2$. The first one that we call weak generation consists in getting ${\\cal Q}_n^K$ by removing $m-n$ disks from ${\\cal Q}_m^K$ and by displacing the $n $ remaining congruent disks which grow continuously and do not overlap.  During a weak generation of ${\\cal Q}_n^K$ from ${\\cal Q}_m^K$, we consider the contact graphs ${\\cal G}(t)$ of the intermediate packings, they represent the contacts disk-disk and disk-boundary. If for each $t$, the contact graph ${\\cal G}(t)$ is isomorphic to the largest common subgraph of the two contact graphs of ${\\cal Q}_n^K$ and ${\\cal Q}_m^K$, we say that the generation is strong. We call strong generator in $K$, an optimal packing ${\\cal Q}_m^K$ which generates strongly all the optimal ${\\cal Q}_k^K$ with $k &lt; m$. We conjecture that if $K$ is compact and convex, there exists an infinite sequence of strong generators in $K$. When $K$ is an equilateral triangle, this conjecture seems to be verified by the sequence of hexagonal packings ${\\cal Q}_{\\Delta (k)}^K$ of $\\Delta (k)=k(k+1)\/2$ disks. In this domain, we also report that up to $n=34$, the Danzer graph of ${\\cal Q}_n^K$ is embedded in the Danzer graph of ${\\cal Q}_{\\Delta (k)}^K$ with $\\Delta (k-1)\\leq n &lt;  \\Delta (k)$. When $K$ is a circle, the first five strong generators appears to be the hexagonal packings defined by Graham and Lubachevsky. When $K$ is a square, we think that our conjecture is verified by a series of packings proposed by Nurmela and al. In the same domain, we give an alternative conjecture by considering another packing pattern.<\/jats:p>","DOI":"10.37236\/113","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:37:03Z","timestamp":1578717423000},"source":"Crossref","is-referenced-by-count":0,"title":["Generation of Optimal Packings from Optimal Packings"],"prefix":"10.37236","volume":"16","author":[{"given":"Thierry","family":"Gensane","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2008,2,20]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r24\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r24\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T03:57:40Z","timestamp":1579319860000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i1r24"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,2,20]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/113","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2008,2,20]]},"article-number":"R24"}}