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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 2150-959X

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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, co-bipartite 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.

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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:	2150-959X
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 fundamentalŉykh issledovaniĭ [Russian Foundation for Basic Research] (RFFI), FAP
Grant number:	11-01-00107-a (RFFI), 2010-1.3.1-111-017-012 (FAP)
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URI:	http://wrap.warwick.ac.uk/id/eprint/38522