{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:15Z","timestamp":1753893855428,"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>In this paper, we prove that in a multigraph whose density $\\Gamma$ exceeds the maxiimum vertex degree $\\Delta$, the collection of minimal clusters (maximally dense sets of vertices) is cycle-free. We also prove that for multigraphs with $\\Gamma\\gt\\Delta+1$, the size of any cluster is bounded from the above by $(\\Gamma-3)\/(\\Gamma-\\Delta-1)$. Finally, we show that two well-known lower bounds for the chromatic index of a multigraph are equal.<\/jats:p>","DOI":"10.37236\/4847","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T23:52:20Z","timestamp":1578700340000},"source":"Crossref","is-referenced-by-count":1,"title":["Clusters in a Multigraph with Elevated Density"],"prefix":"10.37236","volume":"14","author":[{"given":"Mark K.","family":"Goldberg","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2007,1,10]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v14i1r10\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v14i1r10\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:04:58Z","timestamp":1579302298000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v14i1r10"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,1,10]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2007,1,3]]}},"URL":"https:\/\/doi.org\/10.37236\/4847","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2007,1,10]]},"article-number":"R10"}}