TY - JOUR
T1 - The structure of almost all graphs in a hereditary property
AU - Alon, Noga
AU - Balogh, József
AU - Bollobás, Béla
AU - Morris, Robert
N1 - Funding Information:
✩ The first author was supported in part by a USA Israeli BSF grant, by a grant from the Israel Science Foundation, by an ERC advanced grant, by NSF Grant CCF 0832797 and by the Ambrose Monell Foundation. The second author was supported by NSF CAREER Grants DMS-0745185 and DMS-0600303, UIUC Campus Research Board Grants 09072 and 08086, and OTKA Grant K76099. The third author was supported by NSF Grants DMS-0505550, CNS-0721983 and CCF-0728928, and ARO Grant W911NF-06-1-0076. The fourth author was supported by a Research Fellowship from Murray Edwards College, Cambridge, ERC Advanced grant DMMCA, and a JSPS fellowship. E-mail addresses: [email protected] (N. Alon), [email protected] (J. Balogh), [email protected] (B. Bollobás), [email protected] (R. Morris).
PY - 2011/3
Y1 - 2011/3
N2 - A hereditary property of graphs is a collection of graphs which is closed under taking induced subgraphs. The speed of P is the function n→|Pn|, where Pn denotes the graphs of order n in P. It was shown by Alekseev, and by Bollobás and Thomason, that if P is a hereditary property of graphs then. |Pn|=2(1-1/r+o(1))(n2), where r=r(P)j{cyrillic,ukrainian}N is the so-called 'colouring number' of P. However, their results tell us very little about the structure of a typical graph GεP. In this paper we describe the structure of almost every graph in a hereditary property of graphs, P. As a consequence, we derive essentially optimal bounds on the speed of P, improving the Alekseev-Bollobás-Thomason Theorem, and also generalising results of Balogh, Bollobás and Simonovits.
AB - A hereditary property of graphs is a collection of graphs which is closed under taking induced subgraphs. The speed of P is the function n→|Pn|, where Pn denotes the graphs of order n in P. It was shown by Alekseev, and by Bollobás and Thomason, that if P is a hereditary property of graphs then. |Pn|=2(1-1/r+o(1))(n2), where r=r(P)j{cyrillic,ukrainian}N is the so-called 'colouring number' of P. However, their results tell us very little about the structure of a typical graph GεP. In this paper we describe the structure of almost every graph in a hereditary property of graphs, P. As a consequence, we derive essentially optimal bounds on the speed of P, improving the Alekseev-Bollobás-Thomason Theorem, and also generalising results of Balogh, Bollobás and Simonovits.
KW - Entropy
KW - Hereditary property
KW - Structure of graphs
UR - http://www.scopus.com/inward/record.url?scp=78951485718&partnerID=8YFLogxK
U2 - 10.1016/j.jctb.2010.10.001
DO - 10.1016/j.jctb.2010.10.001
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AN - SCOPUS:78951485718
SN - 0095-8956
VL - 101
SP - 85
EP - 110
JO - Journal of Combinatorial Theory. Series B
JF - Journal of Combinatorial Theory. Series B
IS - 2
ER -