{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:53Z","timestamp":1753893833303,"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>Motivated by a question on the maximal number of vertex disjoint Schrijver graphs in the Kneser graph, we investigate the following function, denoted by $f(n,k)$: the maximal number of Hamiltonian cycles on an $n$ element set, such that no two cycles share a common independent set of size more than $k$. We shall mainly be interested in the behavior of $f(n,k)$ when $k$ is a linear function of $n$, namely $k=cn$. We show a threshold phenomenon: there exists a constant $c_t$ such that for $c&lt;c_t$, $f(n,cn)$ is bounded by a constant depending only on $c$ and not on $n$, and for $c_t &lt;c$, $f(n,cn)$ is exponentially large in $n ~(n \\to \\infty)$. We prove that $0.26 &lt; c_t &lt; 0.36$, but the exact value of $c_t$ is not determined. For the lower bound we prove a technical lemma, which for graphs that are the union of two Hamiltonian cycles establishes a relation between the independence number and the number of $K_4$ subgraphs. A corollary of this lemma is that if a graph $G$ on $n&gt;12$ vertices is the union of\u00a0 two Hamiltonian cycles and $\\alpha(G)=n\/4$, then $V(G)$ can be covered by vertex-disjoint $K_4$ subgraphs.<\/jats:p>","DOI":"10.37236\/6493","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T14:51:50Z","timestamp":1578667910000},"source":"Crossref","is-referenced-by-count":0,"title":["Independent Sets in the Union of Two Hamiltonian Cycles"],"prefix":"10.37236","volume":"25","author":[{"given":"Ron","family":"Aharoni","sequence":"first","affiliation":[]},{"given":"Daniel","family":"Solt\u00e9sz","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2018,12,21]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i4p48\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i4p48\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:20:39Z","timestamp":1579234839000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v25i4p48"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,12,21]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2018,10,5]]}},"URL":"https:\/\/doi.org\/10.37236\/6493","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2018,12,21]]},"article-number":"P4.48"}}