20 research outputs found
Hereditary properties of tournaments
A collection of unlabelled tournaments P is called a hereditary property if
it is closed under isomorphism and under taking induced sub-tournaments. The
speed of P is the function n -> |P_n|, where P_n = {T \in P : |V(T)| = n}. In
this paper, we prove that there is a jump in the possible speeds of a
hereditary property of tournaments, from polynomial to exponential speed.
Moreover, we determine the minimal exponential speed, |P_n| = c^(n + o(n)),
where c = 1.47... is the largest real root of the polynomial x^3 = x^2 + 1, and
the unique hereditary property with this speed.Comment: 28 pgs, 2 figures, submitted November 200
Hereditary properties of combinatorial structures: posets and oriented graphs
A hereditary property of combinatorial structures is a collection of
structures (e.g. graphs, posets) which is closed under isomorphism, closed
under taking induced substructures (e.g. induced subgraphs), and contains
arbitrarily large structures. Given a property P, we write P_n for the
collection of distinct (i.e., non-isomorphic) structures in a property P with n
vertices, and call the function n -> |P_n| the speed (or unlabelled speed) of
P. Also, we write P^n for the collection of distinct labelled structures in P
with vertices labelled 1,...,n, and call the function n -> |P^n| the labelled
speed of P.
The possible labelled speeds of a hereditary property of graphs have been
extensively studied, and the aim of this paper is to investigate the possible
speeds of other combinatorial structures, namely posets and oriented graphs.
More precisely, we show that (for sufficiently large n), the labelled speed of
a hereditary property of posets is either 1, or exactly a polynomial, or at
least 2^n - 1. We also show that there is an initial jump in the possible
unlabelled speeds of hereditary properties of posets, tournaments and directed
graphs, from bounded to linear speed, and give a sharp lower bound on the
possible linear speeds in each case.Comment: 26 pgs, no figure