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Graphs and Groups, Cyles and Coverings
Abstrats
Many yle lengths in triangle-free graphs with high hromati number
Alexandr Kostohka
University of Illinois at Urbana-Champaign, USA
Sobolev Institute of Mathematis, Novosibirsk, Russia
kostohkmath.uiu.edu
Erdos onjetured that a trianglefree graph G with hromati number k ≥ k0 (ε) ontains yles of at
least k 2−ε dierent lengths as k → ∞. We prove the stronger fat that every trianglefree graph G with
1
hromati number k ≥ k0 (ε) ontains yles of ( 64
− ε)k 2 log k onseutive lengths, and a yle of length
1
2
at least ( 4 − ε)k log k . Sine there are triangle-free graphs with hromati number k and at most roughly
4k 2 log k verties for large k , these results are tight up to a onstant fator. We also give new lower bounds
on the irumferene and the number of dierent yle lengths for k -hromati graphs in other hereditary
lasses.
This is joint work with B. Sudakov and J. Verstraete.
Akademgorodok, Novosibirsk, Russia
September, 23-26, 2014