{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:39Z","timestamp":1753893819535,"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>Let $x,y\\in (0,1]$, and let $A,B,C$ be disjoint nonempty stable subsets of a graph $G$, where every vertex in $A$ has at least $x|B|$ neighbours in $B$, and every vertex in $B$ has at least $y|C|$ neighbours in $C$, and there are no edges between $A,C$. We denote by $\\phi(x,y)$ the maximum $z$ such that, in all such graphs $G$, there is a vertex $v\\in C$ that is joined to at least $z|A|$ vertices in $A$ by two-edge paths. This function has some interesting properties: we show, for instance, that $\\phi(x,y)=\\phi(y,x)$ for all $x,y$, and there is a discontinuity in $\\phi(x,x)$ when $1\/x$ is an integer. For $z=1\/2, 2\/3,1\/3,3\/4,2\/5,3\/5$, we try to find the (complicated) boundary between the set of pairs $(x,y)$ with $\\phi(x,y)\\ge z$ and the pairs with $\\phi(x,y)&lt;z$. We also consider what happens if in addition every vertex in $B$ has at least $x|A|$ neighbours in $A$, and every vertex in $C$ has at least $y|B|$ neighbours in $B$.\r\nWe raise several questions and conjectures; for instance, it is open whether $\\phi(x,x)\\ge 1\/2$ for all $x&gt;1\/3$.<\/jats:p>","DOI":"10.37236\/8451","type":"journal-article","created":{"date-parts":[[2022,6,3]],"date-time":"2022-06-03T07:21:01Z","timestamp":1654240861000},"source":"Crossref","is-referenced-by-count":0,"title":["Concatenating Bipartite Graphs"],"prefix":"10.37236","volume":"29","author":[{"given":"Maria","family":"Chudnovsky","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Patrick","family":"Hompe","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alex","family":"Scott","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Paul","family":"Seymour","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sophie","family":"Spirkl","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2022,6,3]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i2p47\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i2p47\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,6,3]],"date-time":"2022-06-03T07:21:02Z","timestamp":1654240862000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v29i2p47"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,3]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,4,8]]}},"URL":"https:\/\/doi.org\/10.37236\/8451","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2022,6,3]]},"article-number":"P2.47"}}