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CATEGORIES:Pure Maths Colloquium
SUMMARY:Triangle factors in random graphs - Oliver Riorda
n (Oxford)
DTSTART;TZID=Europe/London:20190225T160000
DTEND;TZID=Europe/London:20190225T170000
UID:TALK119128AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/119128
DESCRIPTION:The Erdös—Rényi or `binomial’ random graph G(n\,p)
consists of n vertices\, with each pair connected
by an edge with probability p\, independently of
the others. The nature of the model means that `lo
cal’ properties (such as individual vertex degrees
) tend to be relatively easy to study\, whereas `g
lobal’ properties (such as the size of the largest
component) are much harder. An interesting class
of questions relates one to the other. For example
\, if p=p(n) is chosen so that G(n\,p) has whp (`w
ith high probability’\, i.e.\, with probability te
nding to 1 as n tends to infinity) minimum degree
at least 1\, does it also have (whp) the global pr
operty of connectedness? The answer is yes\, as sh
own already by Erdös and Rényi in 1960. What about
minimum degree 2 and containing a Hamilton cycle?
Again yes\, as shown by Komlós and Szemerédi in 1
983. What about every vertex being in a triangle\,
and the graph containing a triangle factor\, i.e.
\, a set of n/3 disjoint triangles covering all th
e vertices? This question turned out to be much ha
rder\, and was eventually answered (approximately)
by Johansson\, Kahn and Vu in 2008.\n\nIn this ta
lk I will describe at least some aspects of the pr
oof of the last result\, as well as a related rece
nt development. The aim is not so much to present
particular results\, but rather to give a flavour
of the range of methods that are used in studying
this type of problem.\n\nThe colloquium is followe
d by a wine reception in Central Core.\n\n\n\n
LOCATION:MR3\, CMS
CONTACT:HoD Secretary\, DPMMS
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