{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:49Z","timestamp":1753893829049,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>In this paper we obtain the expectation and variance of the number\u00a0of Euler tours of a random Eulerian directed graph with fixed\u00a0out-degree sequence. We use this to obtain the asymptotic\u00a0distribution of the number of Euler tours of a random $d$-in\/$d$-out\u00a0graph and prove a concentration result. We are then able to show\u00a0that a very simple approach for uniform sampling or approximately\u00a0counting Euler tours yields algorithms running in expected\u00a0polynomial time for almost every $d$-in\/$d$-out graph. We make use\u00a0of the BEST theorem of de Bruijn, van Aardenne-Ehrenfest, Smith and\u00a0Tutte, which shows that the number of Euler tours of an Eulerian\u00a0directed graph with out-degree sequence $\\mathbf{d}$ is the product\u00a0of the number of arborescences and the term $\\frac{1}{|V|}[\\prod_{v\\in V}(d_v-1)!]$. Therefore most of our effort is towards\u00a0estimating the moments of the number of arborescences of a random\u00a0graph with fixed out-degree sequence.<\/jats:p>","DOI":"10.37236\/2377","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T01:24:05Z","timestamp":1578705845000},"source":"Crossref","is-referenced-by-count":0,"title":["The Number of Euler Tours of Random Directed Graphs"],"prefix":"10.37236","volume":"20","author":[{"given":"P\u00e1id\u00ed","family":"Creed","sequence":"first","affiliation":[]},{"given":"Mary","family":"Cryan","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2013,8,9]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i3p13\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i3p13\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:16:18Z","timestamp":1579259778000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i3p13"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,8,9]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2013,7,19]]}},"URL":"https:\/\/doi.org\/10.37236\/2377","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2013,8,9]]},"article-number":"P13"}}