{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:01Z","timestamp":1753893781095,"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 $\\gamma (n,\\delta)$ denote the largest possible domination number for a graph of order $n$ and minimum degree $\\delta$. This paper is concerned with the behavior of the right side of the sequence $$\\gamma (n,0) \\ge \\gamma (n,1) \\ge \\cdots \\ge \\gamma (n,n-1) = 1. $$ We set $ \\delta _k(n) = \\max \\{ \\delta \\, \\vert \\, \\gamma (n,\\delta) \\ge k \\}$, $k \\ge 1.$ Our main result is that for any fixed $k \\ge 2$ there is a constant $c_k$ such that for sufficiently large $n$, $$ n-c_kn^{(k-1)\/k} \\le \\delta _{k+1}(n) \\le n - n^{(k-1)\/k}. $$  The lower bound is obtained by use of circulant graphs.  We also show that for $n$ sufficiently large relative to $k$, $\\gamma (n,\\delta _k(n)) = k$. The case $k=3$ is examined in further detail. The existence of  circulant graphs with domination number greater than 2 is related to a kind of difference set in ${\\bf Z}_n$.<\/jats:p>","DOI":"10.37236\/1536","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T02:02:35Z","timestamp":1578708155000},"source":"Crossref","is-referenced-by-count":0,"title":["Tight Upper Bounds for the Domination Numbers of Graphs with Given Order and Minimum Degree, II"],"prefix":"10.37236","volume":"7","author":[{"given":"W. Edwin","family":"Clark","sequence":"first","affiliation":[]},{"given":"Larry A.","family":"Dunning","sequence":"additional","affiliation":[]},{"given":"Stephen","family":"Suen","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2000,11,6]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v7i1r58\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v7i1r58\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T05:21:01Z","timestamp":1579324861000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v7i1r58"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,11,6]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2000,1,1]]}},"URL":"https:\/\/doi.org\/10.37236\/1536","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2000,11,6]]},"article-number":"R58"}}