{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:38Z","timestamp":1753893818213,"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 present two theorems in the \"discrete differential geometry\" of positively curved spaces. The first is a combinatorial analog of the Bonnet-Myers theorem:  $\\bullet$  A combinatorial 3-manifold whose edges have degree at most five has edge-diameter at most five. When all edges have unit length, this degree bound is equivalent to an angle-deficit along each edge. It is for this reason we call such spaces positively curved. Our second main result is analogous to the sphere theorems of Toponogov and Cheng:  $\\bullet$ A positively curved 3-manifold, as above, in which vertices $v$ and $w$ have edge-distance five is a sphere whose triangulation is completely determined by the structure of $Lk(v)$ or $Lk(w)$. In fact, we provide a procedure for constructing a maximum diameter sphere from a suitable $Lk(v)$ or $Lk(w)$. The compactness of these spaces (without an explicit diameter bound) was first proved via analytic arguments in a 1973 paper by David Stone. Our proof is completely combinatorial, provides sharp bounds, and follows closely the proof strategy for the classical results.<\/jats:p>","DOI":"10.37236\/321","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T23:14:50Z","timestamp":1578698090000},"source":"Crossref","is-referenced-by-count":1,"title":["Positively Curved Combinatorial 3-Manifolds"],"prefix":"10.37236","volume":"17","author":[{"given":"Aaron","family":"Trout","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2010,3,29]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r49\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r49\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T18:57:19Z","timestamp":1579287439000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v17i1r49"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,3,29]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2010,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/321","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2010,3,29]]},"article-number":"R49"}}