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SUMMARY:Raphael Yuster (University of Haifa)
DTSTART:20210621T140000Z
DTEND:20210621T150000Z
DTSTAMP:20260423T021114Z
UID:EPC/58
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/EPC/58/">Ham
 ilton cycles above expectation in r-graphs and quasi-random r-graphs</a>\n
 by Raphael Yuster (University of Haifa) as part of Extremal and probabilis
 tic combinatorics webinar\n\n\nAbstract\nLet $H_r(n\,p)$ denote the maximu
 m number of (tight) Hamilton cycles in an n-vertex r-graph with density $p
  \\in (0\,1)$. The expected number of Hamilton cycles in the random r-grap
 h $G_r(n\,p)$ is clearly $E(n\,p)=p^n(n-1)!/2$ and in the random r-graph $
 G_r(n\,m)$ with $m=p\\binom{n}{r}$ it is\, in fact\, easily shown to be sl
 ightly smaller than $E(n\,p)$.\n\nFor graphs\, $H_2(n\,p)$ it is proved to
  be only larger than $E(n\,p)$ by a polynomial factor and it is an open pr
 oblem whether a quasi-random graph with density p can be larger than $E(n\
 ,p)$ by a polynomial factor.\n\nFor hypergraphs the situation is drastical
 ly different. For all $r \\ge 3$ it is proved that  $H_r(n\,p)$ is larger 
 than $E(n\,p)$ by an exponential factor and\, moreover\, there are quasi-r
 andom r-graphs with density p whose number of Hamilton cycles is larger th
 an $E(n\,p)$ by an exponential factor.\n
LOCATION:https://researchseminars.org/talk/EPC/58/
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