{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:11Z","timestamp":1753893851359,"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>We consider the Erd\u0151s-R\u00e9nyi random graph process, which is a stochastic process that starts with $n$ vertices and no edges, and at each step adds one new edge chosen uniformly at random from the set of missing edges. Let $\\mathcal{G}(n,m)$ be a graph with $m$ edges obtained after $m$ steps of this process. Each edge $e_i$ ($i=1,2,\\ldots, m$) of $\\mathcal{G}(n,m)$ independently chooses precisely $k \\in\\mathbb{N}$ colours, uniformly at random, from a given set of $n-1$ colours (one may view $e_i$ as a multi-edge). We stop the process prematurely at time $M$ when the following two events hold: $\\mathcal{G}(n,M)$ is connected and every colour occurs at least once ($M={n \\choose 2}$ if some colour does not occur before all edges are present; however, this does not happen asymptotically almost surely). The question addressed in this paper is whether $\\mathcal{G}(n,M)$ has a rainbow spanning tree (that is, multicoloured tree on $n$ vertices). Clearly, both properties are necessary for the desired tree to exist.In 1994, Frieze and McKay investigated the case $k=1$ and the answer to this question is \"yes\" (asymptotically almost surely). However, since the sharp threshold for connectivity is $\\frac {n}{2} \\log n$ and the sharp threshold for seeing all the colours is $\\frac{n}{k} \\log n$, the case $k=2$ is of special importance as in this case the two processes keep up with one another. In this paper, we show that asymptotically almost surely the answer is \"yes\" also for $k \\ge 2$.<\/jats:p>","DOI":"10.37236\/4642","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T00:31:59Z","timestamp":1578702719000},"source":"Crossref","is-referenced-by-count":1,"title":["Power of $k$ Choices and Rainbow Spanning Trees in Random Graphs"],"prefix":"10.37236","volume":"22","author":[{"given":"Deepak","family":"Bal","sequence":"first","affiliation":[]},{"given":"Patrick","family":"Bennett","sequence":"additional","affiliation":[]},{"given":"Alan","family":"Frieze","sequence":"additional","affiliation":[]},{"given":"Pawe\u0142","family":"Pra\u0142at","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2015,2,16]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i1p29\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v22i1p29\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T10:37:15Z","timestamp":1579257435000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v22i1p29"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,2,16]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2015,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/4642","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2015,2,16]]},"article-number":"P1.29"}}