{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:29Z","timestamp":1753893869925,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Given a family of graphs $\\mathcal{F}$, we consider the $\\mathcal{F}$-saturation game.\u00a0 In this game, two players alternate adding edges to an initially empty graph on $n$ vertices, with the only constraint being that neither player can add an edge that creates a subgraph that lies in $\\mathcal{F}$.\u00a0 The game ends when no more edges can be added to the graph.\u00a0 One of the players wishes to end the game as quickly as possible, while the other wishes to prolong the game.\u00a0 We let $\\textrm{sat}_g(\\mathcal{F};n)$ denote the number of edges that are in the final graph when both players play optimally.\r\nThe $\\{C_3\\}$-saturation game was the first saturation game to be considered, but as of now the order of magnitude of $\\textrm{sat}_g(\\{C_3\\},n)$ remains unknown.\u00a0 We consider a variation of this game.\u00a0 Let $\\mathcal{C}_{2k+1}:=\\{C_3,\\ C_5,\\ldots,C_{2k+1}\\}$. We prove that $\\textrm{sat}_g(\\mathcal{C}_{2k+1};n)\\ge(\\frac{1}{4}-\\epsilon_k)n^2+o(n^2)$ for all $k\\ge 2$ and that $\\textrm{sat}_g(\\mathcal{C}_{2k+1};n)\\le (\\frac{1}{4}-\\epsilon'_k)n^2+o(n^2)$ for all $k\\ge 4$, with $\\epsilon_k&lt;\\frac{1}{4}$ and $\\epsilon'_k&gt;0$ constants tending to 0 as $k\\to \\infty$.\u00a0 In addition to this we prove $\\textrm{sat}_g(\\{C_{2k+1}\\};n)\\le \\frac{4}{27}n^2+o(n^2)$ for all $k\\ge 2$, and $\\textrm{sat}_g(\\mathcal{C}_\\infty\\setminus C_3;n)\\le 2n-2$, where $\\mathcal{C}_\\infty$ denotes the set of all odd cycles.<\/jats:p>","DOI":"10.37236\/8113","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T06:04:55Z","timestamp":1578636295000},"source":"Crossref","is-referenced-by-count":2,"title":["Saturation Games for Odd Cycles"],"prefix":"10.37236","volume":"26","author":[{"given":"Sam","family":"Spiro","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2019,10,11]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i4p11\/7936","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i4p11\/7936","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:02:47Z","timestamp":1579233767000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v26i4p11"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10,11]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2019,10,11]]}},"URL":"https:\/\/doi.org\/10.37236\/8113","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2019,10,11]]},"article-number":"P4.11"}}