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On Betti numbers of flag complexes with forbidden induced subgraphs

Part of: Algebraic combinatorics Graph theory Low-dimensional topology

Published online by Cambridge University Press:  07 June 2019

KARIM ADIPRASITO ,
ERAN NEVO  and
MARTIN TANCER
KARIM ADIPRASITO*
Affiliation:
Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904 Israel.
ERAN NEVO
Affiliation:
Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, 91904 Israel.
MARTIN TANCER
Affiliation:
Department of Applied Mathematics, Charles University in Prague, Malostranské náměstí 25, 118 00, Praha 1. e-mail: tancer@kam.mff.cuni.cz
*
e-mails: adiprasito@math.huji.ac.il, nevo.eran@gmail.com
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Abstract

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.

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05E45: Combinatorial aspects of simplicial complexes 57M15: Relations with graph theory
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 3 , May 2020 , pp. 567 - 600
Copyright
Copyright © Cambridge Philosophical Society 2019

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Footnotes

†

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 GAČR 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.

References

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