{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T07:48:16Z","timestamp":1780386496060,"version":"3.54.1"},"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>Let $ { k\\ge 2}$ and $ {g\\ge3}$ be integers. A $ {(k,g)}$-graph is a $ {k}$-regular graph with girth  (length of a smallest cycle)  exactly $ {g}$.  A $ {(k,g)}$-cage is a $ {(k,g)}$-graph of minimum order.  Let $ {v(k,g)}$ be the order of a  $ {(k,g)}$-cage. The problem of determining $ {v(k,g)}$ is unsolved for most pairs $ {(k,g)}$ and is extremely hard in the general case. It is easy to establish the following  lower bounds for $ {v(k,g)}$: $ {v(k,g)\\ge} {{k(k-1)^{(g-1)\/2}-2}\\over {k-2}}$  for  $ {g}$ odd, and $ {v(k,g)\\ge}   { {2(k-1)^{g\/2}-2}\\over {k-2}}$  for $ {g}$ even. The best known  upper bounds are roughly the squares of the lower bounds. In this paper we establish general upper bounds on  $ {v(k,g)}$ which are roughly the 3\/2 power of the lower bounds, and we provide explicit constructions for such  $ {(k,g)}$-graphs.<\/jats:p>","DOI":"10.37236\/1328","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T01:55:26Z","timestamp":1578707726000},"source":"Crossref","is-referenced-by-count":15,"title":["Upper Bounds on the Order of Cages"],"prefix":"10.37236","volume":"4","author":[{"given":"F.","family":"Lazebnik","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"V. A.","family":"Ustimenko","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"A. J.","family":"Woldar","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"23455","published-online":{"date-parts":[[1996,11,21]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v4i2r13\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v4i2r13\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T06:15:10Z","timestamp":1579328110000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v4i2r13"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,11,21]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,1,10]]}},"URL":"https:\/\/doi.org\/10.37236\/1328","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,11,21]]},"article-number":"R13"}}