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SUMMARY:Minimum degree stability and locally colourable graphs - Dr F.Illi
 ngworth (Oxford)
DTSTART:20220210T143000Z
DTEND:20220210T153000Z
UID:TALK169943@talks.cam.ac.uk
CONTACT:103978
DESCRIPTION:We tie together two natural but\, a priori\, different themes.
  As a starting\npoint consider Erdős and Simonovits's classical edge stab
 ility for an (r +\n1)-chromatic graph H. This says that any n-vertex H-fre
 e graph with (1 − 1/r +\no(1))*(n choose 2) edges is close to (within o(
 n^2) edges of) r-partite. This\nis false if 1 − 1/r is replaced by any s
 maller constant. However\, instead of\ninsisting on many edges\, what if w
 e ask that the n-vertex graph has large\nminimum degree? This is the basic
  question of minimum degree stability: what\nconstant c guarantees that an
 y n-vertex H-free graph with minimum degree\ngreater than cn is close to r
 -partite? c depends not just on chromatic number\nof H but also on its fin
 er structure.\n\nSomewhat surprisingly\, answering the minimum degree stab
 ility question\nrequires understanding locally colourable graphs -- graphs
  in which every\nneighbourhood has small chromatic number -- with large mi
 nimum degree. This is\na natural local-to-global colouring question: if ev
 ery neighbourhood is big\nand has small chromatic number must the whole gr
 aph have small chromatic\nnumber? The triangle-free case has a rich histor
 y. The more general case has\nsome similarities but also striking differen
 ces.\n
LOCATION:MR5 Centre for Mathematical Sciences
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