{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:08Z","timestamp":1753893788004,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>This work examines the problem of clique enumeration on a graph by exploiting its clique covers. The principle of inclusion\/exclusion is applied to determine the number of cliques of size $r$ in the graph union of a set $\\mathcal{C} = \\{c_1, \\ldots, c_m\\}$ of $m$ cliques. This leads to a deeper examination of the sets involved and to an orbit partition, $\\Gamma$, of the power set $\\mathcal{P}(\\mathcal{N}_{m})$ of $\\mathcal{N}_{m} = \\{1, \\ldots, m\\}$. Applied to the cliques, this partition gives insight into clique enumeration and yields new results on cliques within a clique cover, including expressions for the number of cliques of size $r$ as well as generating functions for the cliques on these graphs. The quotient graph modulo this partition provides a succinct representation to determine cliques and maximal cliques in the graph union. The partition also provides a natural and powerful framework for related problems, such as the enumeration of induced connected components, by drawing upon a connection to extremal set theory through intersecting sets.<\/jats:p>","DOI":"10.37236\/11463","type":"journal-article","created":{"date-parts":[[2023,6,30]],"date-time":"2023-06-30T08:27:05Z","timestamp":1688113625000},"source":"Crossref","is-referenced-by-count":0,"title":["How Many Cliques Can a Clique Cover Cover?"],"prefix":"10.37236","volume":"30","author":[{"given":"Pavel","family":"Shuldiner","sequence":"first","affiliation":[]},{"given":"R. Wayne","family":"Oldford","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2023,6,30]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i2p53\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i2p53\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,30]],"date-time":"2023-06-30T08:27:06Z","timestamp":1688113626000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v30i2p53"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,30]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,4,7]]}},"URL":"https:\/\/doi.org\/10.37236\/11463","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2023,6,30]]},"article-number":"P2.53"}}