Growth constants of minor-closed classes of graphs

Authors:
Olivier Bernardi;Marc Noy;Dominic Welsh
Affiliations:
CNRS, Département de Mathématiques, Université Paris-Sud, 91405 Orsay Cedex, France;Universitat Politècnica de Catalunya, Jordi Girona 1-3, 08034 Barcelona, Spain;University of Oxford, Mathematical Institute, 24-29 St Giles', Oxford OX1 3LB, UK
Venue:
Journal of Combinatorial Theory Series B
Year:
2010

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Abstract

A minor-closed class of graphs is a set of labelled graphs which is closed under isomorphism and under taking minors. For a minor-closed class G, let g"n be the number of graphs in G which have n vertices. The growth constant of G is @c=lim@?sup(g"n/n!)^1^/^n. We study the properties of the set @C of growth constants of minor-closed classes of graphs. Among other results, we show that @C does not contain any number in the interval [0,2], besides 0, 1, @x and 2, where @x~1.76. An infinity of further gaps is found by determining all the possible growth constants between 2 and @d~2.25159. Our results give in fact a complete characterization of all the minor-closed classes with growth constant at most @d in terms of their excluded minors.