We analyse the asymptotic extremal growth rate of the Betti numbers of clique complexes of graphs on n vertices not containing a fixed forbidden induced subgraph H. In particular, we prove a theorem of the alternative: for any H the growth rate achieves exactly one of five possible exponentials, that is, independent of the field of coefficients, the nth root of the maximal total Betti number over n-vertex graphs with no induced copy of H has a limit, as n tends to infinity, and, ranging over all H, exactly five different limits are attained. For the interesting case where H is the 4-cycle, the above limit is 1, and we prove a superpolynomial upper bound.
|Original language||American English|
|Number of pages||34|
|Journal||Mathematical Proceedings of the Cambridge Philosophical Society|
|State||Published - 2020|
Bibliographical noteFunding Information:
† Supported by ERC-2016-STG 716424 - CASe and Israel Science Foundation grant 1050/16. ‡ Partially supported by Israel Science Foundation grants ISF-805/11, ISF-1695/15, by grant 2528/16 of the ISF-NRF Singapore joint research program, and by ISF-BSF joint grant 2016288. § Partially supported by the GACˇ R grant 16-01602Y and by Charles University project UNCE/SCI/004. Part of this work was done when M. T. was affiliated with IST Austria.
Copyright © Cambridge Philosophical Society 2019.
- 2010 Mathematics Subject Classification: 05C35 05E45 57M15