{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:06:48Z","timestamp":1758823608281,"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>Let ${\\cal F}_{n,t_r(n)}$ denote the family  of all graphs on $n$ vertices and $t_r(n)$ edges,  where $t_r(n)$ is the number of edges in the Tur\u00e1n's graph $T_r(n)$ \u2013 the complete $r$-partite graph on $n$ vertices with partition sizes  as equal as possible. For a graph $G$ and a positive integer $\\lambda$, let $P_G(\\lambda)$ denote the number of proper vertex colorings of $G$ with at most $\\lambda$ colors, and let $f(n,t_r(n),\\lambda) = \\max\\{P_G(\\lambda):G \\in {\\cal F}_{n,t_r(n)}\\}$. We prove that for all $n\\ge r\\ge 2$,  $f(n,t_r(n),r+1) = P_{T_r(n)}(r+1)$ and that $T_r(n)$ is the only extremal graph.<\/jats:p>","DOI":"10.37236\/442","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:58:48Z","timestamp":1578715128000},"source":"Crossref","is-referenced-by-count":6,"title":["An Extremal Property of Tur\u00e1n Graphs"],"prefix":"10.37236","volume":"17","author":[{"given":"Felix","family":"Lazebnik","sequence":"first","affiliation":[]},{"given":"Spencer","family":"Tofts","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2010,12,10]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r170\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r170\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:19:07Z","timestamp":1579303147000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v17i1r170"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,12,10]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2010,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/442","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2010,12,10]]},"article-number":"R170"}}