{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:33:02Z","timestamp":1787322782604,"version":"build-2736575974"},"reference-count":3,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2012,1]]},"abstract":"<jats:p>A good edge-labeling of a simple graph is a labeling of its edges with real numbers such that, for any ordered pair of vertices $(u,v)$, there is at most one nondecreasing path from $u$ to $v$. Say a graph is good if it admits a good edge-labeling, and is bad otherwise. Our main result is that any good $n$-vertex graph whose maximum degree is within a constant factor of its average degree (in particular, any good regular graph) has at most $n^{1+o(1)}$ edges. As a corollary, we show that there are bad graphs with arbitrarily large girth, answering a question of Bode, Farzad, and Theis [Good edge-labelings and graphs with girth at least five, preprint, available online at arXiv:1109.1125]. We also prove that for any $\\Delta$, there is a $g$ such that any graph with maximum degree at most $\\Delta$ and girth at least $g$ is good.<\/jats:p>","DOI":"10.1137\/11085414x","type":"journal-article","created":{"date-parts":[[2012,9,11]],"date-time":"2012-09-11T21:13:32Z","timestamp":1347398012000},"page":"1265-1268","source":"Crossref","is-referenced-by-count":2,"title":["On the Density of Nearly Regular Graphs with a Good Edge-Labeling"],"prefix":"10.1137","volume":"26","author":[{"given":"Abbas","family":"Mehrabian","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2012,9,11]]},"reference":[{"key":"atyp_ref1","unstructured":"1.\u2002N. Alon and J. Spencer, <i>The Probabilistic Method<\/i> 3rd ed., John Wiley & Sons Inc., Hoboken, NJ, 2008."},{"key":"atyp_ref3","unstructured":"3.\u2002J.C. Bermond, M. Cosnard, and S. P\u00e9rennes, <i>Directed Acyclic Graphs with Unique Path Property<\/i> Technical report RR-6932, INRIA, Champaign, Illinois, 2009."},{"key":"atyp_ref5","doi-asserted-by":"crossref","first-page":"13","DOI":"10.37236\/1328","volume":"4","author":"Lazebnik V.","year":"1997","journal-title":"Electron. J. Combin."}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/11085414X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:25:45Z","timestamp":1787318745000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/11085414X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,1]]},"references-count":3,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2012,1]]}},"alternative-id":["10.1137\/11085414X"],"URL":"https:\/\/doi.org\/10.1137\/11085414x","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,1]]}}}