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Perfect graphs of fixed density: counting and homogeneous sets

Böttcher, Julia and Taraz, Anusch and Würfl, Andreas (2012) Perfect graphs of fixed density: counting and homogeneous sets. Combinatorics, Probability and Computing, 21 (5). pp. 661-682. ISSN 0963-5483

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Identification Number: 10.1017/S0963548312000181

Abstract

For c ∈ (0,1) let C n(c) denote the set of C n-vertex perfect graphs with density c and let C n(c) denote the set of n-vertex graphs without induced C 5 and with density c. We show that lim n→∞ log 2 |Pmathcal;n(c)|/(n 2) = lim n→∞ log 2|C n(c)|/(n 2 = h(c) with h(c) = 1/2 if 1/4 ≤ c ≤ 3/4 and h(c) = 1/2H(|2c - 1|)otherwise, where H is the binary entropy function. Further, we use this result to deduce that almost all graphs in n(c) have homogeneous sets of linear size. This answers a question raised by Loebl and co-workers.

Item Type: Article
Official URL: http://journals.cambridge.org/action/displayJourna...
Additional Information: © 2012 Cambridge University Press
Subjects: Q Science > QA Mathematics
Sets: Departments > Mathematics
Date Deposited: 23 Aug 2012 08:53
Last Modified: 23 Aug 2012 08:53
URI: http://eprints.lse.ac.uk/id/eprint/45495

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