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SUMMARY:Triangle factors in random graphs - Oliver Riordan\, Oxford
DTSTART:20190225T160000Z
DTEND:20190225T170000Z
UID:TALK115876@talks.cam.ac.uk
CONTACT:Ivan Smith
DESCRIPTION:The Erdös--Rényi or `binomial' random graph G(n\,p) consists
  of n vertices\,\nwith each pair connected by an edge with probability p\,
  independently of the others. The nature of\nthe model means that `local' 
 properties (such as individual vertex degrees)\ntend to be relatively easy
  to study\, whereas `global' properties (such as the size\nof the largest 
 component) are much harder. An interesting class of questions\nrelates one
  to the other. For example\, if p=p(n) is chosen so that G(n\,p)\nhas whp 
 (`with high probability'\, i.e.\, with probability tending to 1 as n tends
  to infinity) minimum\ndegree at least 1\, does it also have (whp) the glo
 bal property of connectedness?\nThe answer is yes\, as shown already by Er
 dös and Rényi in 1960. What about\nminimum degree 2 and containing a Ham
 ilton cycle? Again yes\, as shown by\nKomlós and Szemerédi in 1983. What
  about every vertex being in a triangle\, and the\ngraph containing a tria
 ngle factor\, i.e.\, a set of n/3 disjoint triangles covering\nall the ver
 tices? This question turned out to be much harder\, and was eventually ans
 wered\n(approximately) by Johansson\, Kahn and Vu in 2008.\n\nIn this talk
  I will describe at least some aspects of the proof of the last result\, a
 s\nwell as a related recent development. The aim is not so much to present
  particular\nresults\, but rather to give a flavour of the range of method
 s that are used in studying\nthis type of problem.
LOCATION:MR3
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