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The number of Euler tours of a random $d$-in/$d$-out graph
Abstract : In this paper we obtain the expectation and variance of the number of Euler tours of a random $d$-in/$d$-out directed graph, for $d \geq 2$. We use this to obtain the asymptotic distribution and prove a concentration result. We are then able to show that a very simple approach for uniform sampling or approximately counting Euler tours yields algorithms running in expected polynomial time for almost every $d$-in/$d$-out graph. We make use of the BEST theorem of de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte, which shows that the number of Euler tours of a $d$-in/$d$-out graph is the product of the number of arborescences and the term $[(d-1)!]^n/n$. Therefore most of our effort is towards estimating the asymptotic distribution of the number of arborescences of a random $d$-in/$d$-out graph.
Cited literature [17 references]
https://hal.inria.fr/hal-01185585
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Submitted on : Thursday, August 20, 2015 - 4:33:14 PM
Last modification on : Wednesday, November 24, 2021 - 9:54:07 AM
Long-term archiving on: : Wednesday, April 26, 2017 - 9:57:25 AM
Contributor : Coordination Episciences Iam Connect in order to contact the contributor
Submitted on : Thursday, August 20, 2015 - 4:33:14 PM
Last modification on : Wednesday, November 24, 2021 - 9:54:07 AM
Long-term archiving on: : Wednesday, April 26, 2017 - 9:57:25 AM
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