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CATEGORIES:Combinatorics Seminar
SUMMARY:Minimum degree stability and locally colourable gr
aphs - Dr F.Illingworth (Oxford)
DTSTART;TZID=Europe/London:20220210T143000
DTEND;TZID=Europe/London:20220210T153000
UID:TALK169943AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/169943
DESCRIPTION:We tie together two natural but\, a priori\, diffe
rent themes. As a starting\npoint consider Erdős a
nd Simonovits's classical edge stability for an (r
+\n1)-chromatic graph H. This says that any n-ver
tex H-free graph with (1 − 1/r +\no(1))*(n choose
2) edges is close to (within o(n^2) edges of) r-pa
rtite. This\nis false if 1 − 1/r is replaced by an
y smaller constant. However\, instead of\ninsistin
g on many edges\, what if we ask that the n-vertex
graph has large\nminimum degree? This is the basi
c question of minimum degree stability: what\ncons
tant c guarantees that any n-vertex H-free graph w
ith minimum degree\ngreater than cn is close to r-
partite? c depends not just on chromatic number\no
f H but also on its finer structure.\n\nSomewhat s
urprisingly\, answering the minimum degree stabili
ty question\nrequires understanding locally colour
able graphs -- graphs in which every\nneighbourhoo
d has small chromatic number -- with large minimum
degree. This is\na natural local-to-global colour
ing question: if every neighbourhood is big\nand h
as small chromatic number must the whole graph hav
e small chromatic\nnumber? The triangle-free case
has a rich history. The more general case has\nsom
e similarities but also striking differences.\n
LOCATION:MR5 Centre for Mathematical Sciences
CONTACT:
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