е рщ щ п п р ь × р ьж р п ¹ ж ж т × ш ь жся ь рыя ж п э р ж цсвьс свьс
advertisement
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
