{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:00Z","timestamp":1753893840817,"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>Consider a graph $G$ with chromatic number $k$ and a collection of complete bipartite graphs, or bicliques, that cover the edges of $G$. We prove the following two results:   $\\bullet$ If the bipartite graphs form a partition of the edges of $G$, then their number is at least $2^{\\sqrt{\\log_2 k}}$. This is the first improvement of the easy lower bound of $\\log_2 k$, while the Alon-Saks-Seymour conjecture states that this can be improved to $k-1$.  $\\bullet$ The sum of the orders of the bipartite graphs in the cover is at least $(1-o(1))k\\log_2 k$. This generalizes, in asymptotic form, a result of Katona and Szemer\u00e9di who proved that the minimum is $k\\log_2 k$ when $G$ is a clique.<\/jats:p>","DOI":"10.37236\/272","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T23:23:29Z","timestamp":1578698609000},"source":"Crossref","is-referenced-by-count":5,"title":["Bipartite Coverings and the Chromatic Number"],"prefix":"10.37236","volume":"16","author":[{"given":"Dhruv","family":"Mubayi","sequence":"first","affiliation":[]},{"given":"Sundar","family":"Vishwanathan","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2009,11,30]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1n34\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1n34\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T19:03:32Z","timestamp":1579287812000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i1n34"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,11,30]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/272","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2009,11,30]]},"article-number":"N34"}}