{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,29]],"date-time":"2025-10-29T19:37:39Z","timestamp":1761766659247,"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>If $\\mathcal{C}$ is a clutter with $n$ vertices and $q$ edges whose clutter matrix has column vectors ${\\mathcal A} = \\{v_1, \\ldots, v_q\\}$, we call $\\mathcal{C}$ an Ehrhart clutter if $\\{(v_1,1),\\ldots,(v_q,1)\\} \\subset \\{ 0,1 \\}^{n+1}$ is a Hilbert basis.  Letting $A(P)$ be the Ehrhart ring of $P={\\rm conv}(\\mathcal{A})$, we are able to show that if $\\mathcal{C}$ is a uniform unmixed MFMC clutter, then $\\mathcal{C}$ is an Ehrhart clutter and in this case we provide sharp upper bounds on the Castelnuovo-Mumford regularity and the $a$-invariant of $A(P)$. Motivated by the Conforti-Cornu\u00e9jols conjecture on packing problems, we conjecture that if $\\mathcal{C}$ is both ideal and the clique clutter of a perfect graph, then $\\mathcal{C}$ has the MFMC property. We prove this conjecture for Meyniel graphs by showing that the clique clutters of Meyniel graphs are Ehrhart clutters. In much the same spirit, we provide a simple proof of our conjecture when $\\mathcal{C}$ is a uniform clique clutter of a perfect graph. We close with a generalization of Ehrhart clutters as it relates to total dual integrality.<\/jats:p>","DOI":"10.37236\/324","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:14:42Z","timestamp":1578716082000},"source":"Crossref","is-referenced-by-count":6,"title":["Ehrhart Clutters: Regularity and Max-Flow Min-Cut"],"prefix":"10.37236","volume":"17","author":[{"given":"Jos\u00e9","family":"Mart\u00ednez-Bernal","sequence":"first","affiliation":[]},{"given":"Edwin","family":"O'Shea","sequence":"additional","affiliation":[]},{"given":"Rafael H.","family":"Villarreal","sequence":"additional","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\/v17i1r52\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r52\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:57:12Z","timestamp":1579305432000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v17i1r52"}},"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\/324","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2010,3,29]]},"article-number":"R52"}}