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A note on the speed of hereditary graph properties
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Lozin, Vadim V., Mayhill, Colin and Zamaraev, Victor. (2011) A note on the speed of hereditary graph properties. The Electronic Journal of Combinatorics, Vol.18 (No.1). p. 157. ISSN 2150959X

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Official URL: http://www.combinatorics.org/ojs/index.php/eljc/ar...
Abstract
For a graph property X, let X(n) be the number of graphs with vertex set {1, ..., n} having property X, also known as the speed of X. A property X is called factorial if X is hereditary (i.e. closed under taking induced subgraphs) and n(c1n) <= X(n) <= n(c2n) for some positive constants c(1) and c(2). Hereditary properties with the speed slower than factorial are surprisingly well structured. The situation with factorial properties is more complicated and less explored, although this family includes many properties of theoretical or practical importance, such as planar graphs or graphs of bounded vertex degree. To simplify the study of factorial properties,we propose the following conjecture: the speed of a hereditary property X is factorial if and only if the fastest of the following three properties is factorial: bipartite graphs in X, cobipartite graphs in X and split graphs in X. In this note, we verify the conjecture for hereditary properties defined by forbidden induced subgraphs with at most 4 vertices.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Mathematics 
Library of Congress Subject Headings (LCSH):  Graph theory 
Journal or Publication Title:  The Electronic Journal of Combinatorics 
Publisher:  Electronic Journal of Combinatorics 
ISSN:  2150959X 
Official Date:  August 2011 
Volume:  Vol.18 
Number:  No.1 
Number of Pages:  14 
Page Range:  p. 157 
Status:  Peer Reviewed 
Publication Status:  Published 
Funder:  University of Warwick. Centre for Discrete Mathematics and Its Applications, Rossiĭskiĭ fond fundamentalʹnykh issledovaniĭ [Russian Foundation for Basic Research] (RFBR), FAP 
Grant number:  110100107a (RFFI), 20101.3.1111017012 (FAP) 
References:  [1] V.E. Alekseev, Range of values of entropy of hereditary classes of graphs, Diskret. 
URI:  http://wrap.warwick.ac.uk/id/eprint/38522 
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